11 Lecture Ppt

7232019 11 Lecture Ppt

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VECTOR MECHANICS FOR ENGINEERS

DYNAMICS

Tenth Editionin SI Units

Ferdinand P Beer

E Russell Johnston Jr

Phillip J Cornell

eture NotesBrian P Sel$ Cali$ornia Poltehni State Uniampersit

Sanjeev SanghiIndian Institute o$ Tehnolo Delhi

CHAPTER

copy 2013 The McGraw-Hill Comaie Ic$ All rihamp reere($

11 Kinematics of Particles



copy 2013 The McGraw-Hill Companies Inc All rights reserve

Vetor Mehanis $or Enineers Dna(isT en t h

E d i t i on

i n

S I Uni t s

Contents

)) +

Introduction

Rectilinear Motion Position

elocit Acceleration

etermination of the Motion of aParticle

Sam$le Prolem ampamp(

Sam$le Prolem ampamp)

niform Rectilinear+Motion

niforml Accelerated Rectilinear+

Motion

Motion of Several Particles

Relative Motion

Sam$le Prolem ampamp


e$endent Motion

Sam$le Prolem ampamp-

ra$hical Solution of Rectilinear+

Motion Prolems

ther ra$hical MethodsCurvilinear Motion Position elocit

Acceleration

erivatives of ector 0unctions

Rectangular Com$onents of elocit

and Acceleration

Motion Relative to a 0rame in

TranslationTangential and 1ormal Com$onents

Radial and Transverse Com$onents

Sam$le Prolem ampampamp2

Sam$le Prolem ampampamp(





E d i t i on

i n

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))

Kinematic relationships are used to

help us determine the trajectory of a

golf ball the orbital speed of a

satellite and the accelerations

during acrobatic flying


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E d i t i on

i n

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Introduction

)) -

bull Dynamics includes

Kinetics study of the relations existing between the forces acting on

a body the mass of the body and the motion of the body Kinetics is

used to predict the motion caused by given forces or to determine theforces required to produce a given motion

Kinematics study of the geometry of motion

Relates displacement velocity acceleration and time without reference

to the cause of motionFthrust

Flift

Fdrag




E d i t i on

i n

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Introduction

))

bull Particle kinetics includes

bull Rectilinear motion position velocity and acceleration of a

particle as it moves along a straight line

bull Curvilinear motion position velocity and acceleration of a

particle as it moves along a curved line in two or three

dimensions




E d i t i on

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Rectilinear Motion Position elocit Acceleration

))

bull Rectilinear motion particle moving

along a straight line

bull Position coordinate defined by

positive or negative distance from a

fixed origin on the line

bull The motion of a particle is known if

the position coordinate for particle is

known for every value of time t

bull ay be expressed in the form of a

function eg t t x

or in the form of a graph x vs t




E d i t i on

i n

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)) 0

bull $nstantaneous velocity may be positive or

negative agnitude of velocity is referred

to as par ticle s peed

bull onsider particle which occupies position P at time t and P 991257 at t amp983108t

t

xv

t

x

t

lim

Averag e velocity

Inst ant aneous velocity

bull (rom the definition of a derivative

dt dx

t xv

t

lim

eg

)

t t dt dxv

t t x

T





E d i t i on

in

S I Uni t s


)) 1

bull onsider particle with velocity v at time t andv991257 at t amp983108t

Inst ant aneous accelerationt

va

t

lim

t dt dva

t t v

dt xd

dt dv

t va

t

)

)eg

lim


bull $nstantaneous acceleration may be

positive increasing positive velocity

or decreasing negative velocity

negative decreasing positive velocity

or increasing negative velocity

T i





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2

What is true about the kinematics of a particle

a+ The velocity of a particle is always positive

b+ The velocity of a particle is equal to the slope of

the positiontime graphc+ $f the position of a particle is ero then the

velocity must ero

d+ $f the velocity of a particle is ero then its

acceleration must be ero

T i





E d i t i on

n

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)) )3

bull (rom our example

t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -

at t - s x - xma x - m v - a - ) ms

bull 0hat are x v and a at t - s 1

bull 2ote that vma x occurs when a- and that the

slope of the velocity curve is ero at this point


V M h i $ E i D iT i n





E d i t i on

n

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etermination of the Motion of a Particle

)) ))

bull We often determine accelerations from the forces applied

(kinetics will be covered later)

bull enerally have three classes of motion

acceleration given as a function of time a - f 3t +

acceleration given as a function of position a - f3 x+ acceleration given as a function of velocity a - f3v+

bull an you think of a physical eample of when force is a

function of position When force is a function of velocity

a spring drag

V t M h i $ E i D iT i n





E d i t i on

n

S I Uni t s

Acceleration as a function of time $osition or velocit

)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt

If Kinematic relationship Integrate

V t M h i $ E i D iT e

i n




Vetor Mehanis $or Enineers Dna(isen t h

E d i t i on

n

S I Uni t s


)) )

Determine

bull velocity and elevation above ground attime t

bull highest elevation reached by ball and

corresponding time and

bull time when ball will hit the ground andcorresponding velocity

4all tossed with ) ms vertical velocityfrom window m above ground

5678T$62

bull $ntegrate twice to find v3t + and y3t +

bull 5olve for t when velocity equals ero

3time for maximum elevation+ and

evaluate corresponding altitude

bull 5olve for t when altitude equals ero

3time for ground impact+ and evaluatecorresponding velocity


i n





E d i t i on

n

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)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y

5678T$62bull $ntegrate twice to find v3t + and y3t +


i n





E d i t i on

S I Uni t s


)) )

bull 5olve for t when velocity equals ero and evaluatecorresponding altitude

s

m8)9

s

m)

t t v

s)9)t

bull 5olve for t when altitude equals ero and evaluatecorresponding velocity

s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y

Vetor Mehanis $or Enineers Dna(isT e

i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n




Vetor Mehanis $or Enineers Dna(isn t h

E d i t i on

S I Uni t s


)) )0

4rake mechanism used to reduce gun

recoil consists of piston attached to barrelmoving in fixed cylinder filled with oil

s barrel recoils with initial velocity v0

piston moves and oil is forced through

orifices in piston causing piston and

cylinder to decelerate at rate proportional

to their velocity

Determine v3t + x3t + and v3 x+

kva

5678T$62

bull $ntegrate a - dvdt - kv to find v3t +

bull $ntegrate v3t + - dxdt to find x3t +

bull $ntegrate a - v dvdx - kv to find

v3 x+

Vetor Mehanis $or Enineers Dna(isT en

i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2

bull $ntegrate a - v dvdx - kv to find v3 x+

kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3

bowling ball is dropped from a boat so that it

strikes the surface of a lake with a speed of ms

ssuming the ball experiences a downward

acceleration of a =10 - 001v2

when in the waterdetermine the velocity of the ball when it strikes the

bottom of the lake

Which integral should you choose

v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)

When will the bowling ball start slowing down



ssuming the ball experiences a downwardacceleration of a =10 - 001v2 when in the water

determine the velocity of the ball when it strikes the

bottom of the lake

he velocity would have to be high

enough for the ampamp v term to be bigger

than amp

$y


i n

S




Vetor Mehanis $or Enineers Dna(is t h

E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62

bull Determine the proper kinematic

relationship to apply 3is acceleration

a function of time velocity or

position1

bull Determine the total distance the car

travels in onehalf lap

bull $ntegrate to determine the velocity

after onehalf lap

The car starts from rest and accelerates

according to the relationship

)a v

$t travels around a circular track that has

a radius of meters alculate the

velocity of the car after it has travelled

halfway around the track 0hat is thecar 991257s maximum possible speed1

Vetor Mehanis $or Enineers Dna(isT en t

i n

S




Vetor Mehanis $or Enineers Dna(isth

E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v

hoose the proper kinematic relationship

ltiven )a v

vo - r - m

(ind v after = lap

aximum speed

cceleration is a function of velocity and

we also can determine distance Time is not

involved in the problem so we choose

Determine total distance travelled

)3+ 8 m x r


i n

S





E d i t i on

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rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v

Determine the full integral including limits

8

)

v

vdx dvv

valuate the interval and solve for v

)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e

ake the eponential of each side


i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv

ow do you determine the maimum speed the car can reach

)a v gtelocity is a maximum when

acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I




Vetor Mehanis $or Enineers Dna(ish

E d i t i on

Uni t s

niform Rectilinear Motion

)) +

For a particle in uniform

rectilinear motion the

acceleration is +ero and

the velocity is constant

vt x xvt x x

dt vdx

vdt

dx

t x

x

constant

During freefall a parachutist

reaches terminal velocity when

her weight e-uals the drag

force If motion is in a straight

line this is uniform rectilinear

motion

areful these only apply to

uniform rectilinear motionA


i n

S I





E d i t i on

Uni t s

niforml Accelerated Rectilinear Motion

)) +0

If forces applied to a body

are constant (and in a

constant direction) thenyou have uniformly

accelerated rectilinear

motion

nother eample is free

fall when drag is negligible


i n

S I




h

E d i t i on

Uni t s


)) +1

For a particle in uniformly accelerated rectilinear motion the

acceleration of the particle is constant ou may recogni+e these

constant acceleration e-uations from your physics courses

constantv t

v

dv a dv a dt v v at dt

)

x t

x

dx v at dx v at dt x x v t at dt

constant

v x

v x

dvv a v dv a dx v v a x xdx

areful 0 these only apply to uniformlyaccelerated rectilinear motion1


i n

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h

E d i t i on

Uni t s


)) +2

We may be interested in the motion of several different particleswhose motion may be independent or linked together


i n

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h

E d i t i on

Uni t s

Motion of Several Particles Relative Motion

)) 3

bull (or particles moving along the same line timeshould be recorded from the same starting

instant and displacements should be measured

from the same origin in the same direction

A B A B x x x relative position of B

with respect to A A B A B x x x

A B A B vvv relative velocity of B

with respect to A

A B A B

vvv

A B A B aaa relative acceleration of B

with respect to A A B A B aaa


i n

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E d i t i on

Uni t s


)) )

4all thrown vertically from ) m level

in elevator shaft with initial velocity of

)8 ms t same instant openplatform

elevator passes m level movingupward at ms

Determine ( a) when and where ball hits

elevator and (b ) relative velocity of ball

and elevator at contact

5678T$62bull 5ubstitute initial position and velocity

and constant acceleration of ball into

general equations for uniformly

accelerated rectilinear motion

bull 5ubstitute initial position and constant

velocity of elevator into equation for

uniform rectilinear motion

bull 0rite equation for relative position of

ball with respect to elevator and solve

for ero relative position ie impact

bull 5ubstitute impact time into equation

for position of elevator and relative

velocity of ball with respect to

elevator


i n

S I U




E d i t i on

Uni t s


)) +

5678T$62bull 5ubstitute initial position and velocity and constant

acceleration of ball into general equations for

uniformly accelerated rectilinear motion

)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B

bull 5ubstitute initial position and constant velocity of

elevator into equation for uniform rectilinear motion

t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))

bull 0rite equation for relative position of ball with respect toelevator and solve for ero relative position ie impact

9)8) t t t y E B

s

smeaningles s9

t

t

bull 5ubstitute impact time into equations for position of elevator

and relative velocity of ball with respect to elevator

E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s

Motion of Several Particles e$endent Motion

)) -

bull Bosition of a particle may de pend on position of one

or more other particles

bull Bosition of block B depends on position of block A

5ince rope is of constant length it follows that sum oflengths of segments must be constant

B A x x constant 3one degree of freedom+

bull Bositions of three blocks are dependent C B A x x x constant 3two degrees of freedom+

bull (or linearly related positions similar relations hold

between velocities and accelerations

or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))

Bulley D is attached to a collar whichis pulled down at mms t t -

collar A starts moving down from K

with constant acceleration and ero

initial velocity Knowing that velocityof collar A is mms as it passes L

determine the change in elevation

velocity and acceleration of block B

when block A is at L

5678T$62bull Define origin at upper horiontal surface

with positive displacement downward

bull ollar A has uniformly acceleratedrectilinear motion 5olve for acceleration

and time t to reach L

bull Bulley D has uniform rectilinear motion

alculate change of position at time t

bull 4lock B motion is dependent on motions

of collar A and pulley D 0rite motion

relationship and solve for change of block B position at time t

bull Differentiate motion relation twice to

develop equations for velocity and

acceleration of block B


E

i n

S I Un




E d i t i on

ni t s


))

5678T$62bull Define origin at upper horiontal surface with

positive displacement downward

bull ollar A has uniformly accelerated rectilinearmotion 5olve for acceleration and time t to reach L

5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0

bull Bulley D has uniform rectilinear motion alculatechange of position at time t

x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm

bull 4lock B motion is dependent on motions of collar

A and pulley D 0rite motion relationship and

solve for change of block B position at time t

Total length of cable remains constant

mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1

bull Differentiate motion relation twice to developequations for velocity and acceleration of block B

constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2

Slider block A moves to the left with a

constant velocity of 2 m3s Determine the

velocity of block B

Solution steps

9 Sketch your system and choose

coordinate system

9 Write out constraint e-uation

9 Differentiate the constraint e-uation to

get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3

iven v4 2 m3s left Find v5

x

y4

This length is constant no

matter how the blocks move

Sketch your system and choose coordinates

Differentiate the constraint e-uation to

get velocity

const nts a A B x y L

Define your constraint e-uation(s)

ms amp Bv

ms B

v

2ote that as x A gets bigger y B gets smaller


E d

i n

S I Uni




di t i on

it s

ra$hical Solution of Rectilinear+Motion Prolems

)) -)

ngineers often collect position velocity and acceleration

data raphical solutions are often useful in analy+ing

these dataData Fideltit 4 5ihest Reorded Punh

2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2

ltlt ltltlt ltlt ltlt= lt ltamp

Ti(e 6s7

A e l e r a t i o n 6 7 cceleration datafrom a head impact

during a round of

boing


E d

i n

S I Uni

hi l S l i f R ili M i P l




di t i on

ts


)) -+

bull iven the x-t curve the v-t curve is

e-ual to the x-t curve slope

bull iven the v-t curve the a-t curve is

e-ual to the v-t curve slope


E d

i n

S I Uni t

hi l S l ti f R tili M ti P l




di t i on

ts


)) -

bull iven the a-t curve the change in velocity between t 1 and t 2 is

e-ual to the area under the a-t curve between t 1 and t 2

bull iven the v-t curve the change in position between t 1 and t 2 is

e-ual to the area under the v-t curve between t 1 and t 2


E d

i n

S I Uni t

th hi l M th d




i t i on

ts

ther ra$hical Methods

)) --

bull M oment-ar ea method to determine particle position attime t directly from the a-t curve

)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v

dt at t first moment of area under a-t curve

with respect to t - t 1 line

C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -

bull Method to determine particle acceleration from v-x curve

BC

AB

dx

dv

va

tan

subnor mal to v-x curve


E d i

i n

S I Uni t s

Curvilinear Motion Position elocit Acceleration




ti on

s


)) -

he softball and the car both undergo

curvilinear motion

bull particle moving along a curve other than a

straight line is in cur vilinear motion


E d i t

i n

S I Uni t s





ti on

s


)) -0

bull The position vector of a particle at time t is defined by a vector between

origin of a fixed reference frame and the position occupied by particle

bull onsider a particle which occupies position P defined by at time t and P 991257 defined by at t 983108t r

r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt

$nstantaneous velocity

3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r

instantaneous acceleration 3vector+

bull onsider velocity of a particle at time t and velocity at t 983108t v

v

bull $n general the acceleration vector is not tangentto the particle path and velocity vector


E d i t i

i n

S I Uni t s





i on erivatives of ector 0unctions

)) 3

u P

bull 7et be a vector function of scalar variable u

u

u P uu P

u

P

du

P d

uu

limlim

bull Derivative of vector sum

du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d

bull Derivative of product of scalar and vector functions

bull Derivative of scal ar pr oduct and vector pr oduct

du

d

P du

P d

du

P d

du

d P

du

P d

du

P d

Delete or put in 991260bonus991261 slides


E d i t i

i n

S I Uni t s

Rectangular Com$onents of elocit Acceleration




on Rectangular Com$onents of elocit Acceleration

)) )

bull 0hen position vector of particle P is given by itsrectangular components

k $ yi xr

bull gtelocity vector

k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x

bull cceleration vector

k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +

bull Rectangular components particularly effectivewhen component accelerations can be integrated

independently eg motion of a proCectile

$ a g ya xa $ y x

with initial conditions

$ y x vvv $ y x

$ntegrating twice yields

)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x

bull otion in horiontal direction is uniform

bull otion in vertical direction is uniformly accelerated

bull otion of proCectile could be replaced by two

independent rectilinear motions


E d i t i o

i nS I Uni t s

Sam$le Prolem ampamp lt




on Sam$le Prolem ampamplt

))

proCectile is fired from the edgeof a )m cliff with an initial

velocity of )8 ms at an angle of 983216with the horiontal 2eglecting

air resistance find 3a+ the horiontaldistance from the gun to the point

where the proCectile strikes the

ground 3b+ the greatest elevation

above the ground reached by the proCectile

5678T$62

bull onsider the vertical and horiontal motion

separately 3they are independent+

bull pply equations of motion in ydirection

bull pply equations of motion in xdirection

bull Determine time t for proCectile to hit the

ground use this to find the horiontal

distance

bull aximum elevation occurs when v y=0


E d i t i o

i nS I Uni t s






)) -

SOLUION

ltiven 5 6o 7amp2 mgts 56o 7amp-2 m

5a6 7 + =amp mgts( 5a6 7 2 mgts(

6ertical motion 0 uniformly accelerated

ori+ontal motion 0 uniformly accelerated

hoose positive x to the right as shown


E d i t i o

i nS I Uni t s

Sam$le Prolem ampamplt





))

SOLUION

ori+ontal distance

BroCectile strikes the ground at

5olving for t we take the positive root

aximum elevation occurs when v y=0

5ubstitute into equation 3)+ above

5ubstitute t into equation 3+

aximum elevation above the ground -


E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))

If you fire a projectile from 7ampmeters above the ground (see

Problem 8) what launch

angle will give you the greatesthori+ontal distance x

a+ launch angle of

b+ launch angle less than

c+ launch angle greater than

d+ $t depends on the launch velocity


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0

baseball pitching machine991260 throws 991261 baseballs with a

horiontal velocity v $f you

want the height h to be ) m

determine the value of v

5678T$62

bull onsider the vertical and horiontal motionseparately 3they are independent+



bull Determine time t for proCectile to fall to

) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1

naly+e the motion in

the ydirection

ltiven x- ) m yo - ) myf - ) m

(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s

Motion Relative to a 0rame in Translation




n

)) 2

soccer player must consider

the relative motion of the ball

and her teammates when

making a pass

It is critical for a pilot toknow the relative motion

of his aircraft with respect

to the aircraft carrier to

make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D

esignate one frame as the f ixed frame o f r e f er encell other frames not rigidly attached to the fixed

reference frame are movin g frames o f r e f er ence

bull Bosition vectors for particles A and B with respect to

the fixed frame of reference xy$ are and B A r r

bull gtector Coining A and B defines the position of B with respect to the moving frame Ax991257 y991257 $ 991257 and

A Br

A B A B r r r

bull Differentiating twice

A Bv

velocity of B relative to A A B A B vvv

A Ba

acceleration of B relativeto A

A B A B aaa

bull bsolute motion of B can be obtained by combining

motion of A with relative motion of B with respect to

moving reference frame attached to A


E d i t i on

i nS I Uni t s

Sam$le Prolem ampamp=



copy 2013 The McGraw-Hill Companies Inc All rights reserve )) )

utomobile A is traveling east at theconstant speed of kmh s

automobile A crosses the intersection

shown automobile B starts from rest

m north of the intersection andmoves south with a constant

acceleration of ) ms Determine

the position velocity and

acceleration of B relative to A safter A crosses the intersection

5678T$62

bull Define inertial axes for the system

bull Determine the position speed and

acceleration of car at t - s

bull 8sing vectors 3Dqs ))) )) and))+ or a graphical approach determine

the relative position velocity and

acceleration


acceleration of car 4 at t - s


E d i t i on

i nS I Uni t s




copy 2013 The McGraw-Hill Companies Inc All rights reserve )) +

5678T$62 bull Define axes along the road

ltiven v A- kmh a A- 3 x A+ -

3v B+- a B- ) ms 3y A+ - m

Determine motion of utomobile

0e have uniform motion for so

t t - s


E d i t i on

i nS I Uni t s




copy 2013 The McGraw-Hill Companies Inc All rights reserve ))

5678T$62

Determine motion of utomobile 4

0e have uniform acceleration for 4 so

t t - s


E d i t i on

i nS I Uni t s




copy 2013 The McGraw-Hill Companies Inc All rights reserve )) -

5678T$62

0e can solve the problems geometrically and apply the arctangent relationship

6r we can solve the problems using vectors to obtain equivalent results

5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j

Bhysically a rider in car would 991260see991261 car 4 travelling south and west


E d i t i on

i nS I Uni t s

Conce$t 3ui4




If you are sitting in train5 looking out the window

it which direction does it

appear that train ismoving

a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s

Tangential and 1ormal Com$onents




If we have an idea of the path of a vehicle it is often convenient

to analy+e the motion using tangential and normal components

(sometimes called path coordinates)


E d i t i on

i nS I Uni t s




copy 2013 The McGraw-Hill Companies Inc All rights reserve )) 0

bull The tangential direction 3et+ is tangent to the path of the particle This velocity vector of a particle is in this direction

x

y

et

en

bull The normal direction 3en+ is perpendicular to et and points

towards the inside of the curve

v4 vt et

983154- the instantaneous

radius of curvature

dv v

dt

t n

a e e

v tv e

bull The acceleration can have components in both the en and et directions


E d i t i on

i nS I Uni t s





bull To derive the acceleration vector in tangentialand normal components define the motion of a

particle as shown in the figure

bull are tangential unit vectors for the particle path at P and P 991257 0hen drawn with

respect to the same origin and

is the angle between them

t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv

bull 0ith the velocity vector expressed as

the particle acceleration may be written as

dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt

bull The tangential component of acceleration

reflects change of speed and the normal

component reflects change of direction

bull The tangential component may be positive or

negative 2ormal component always points

toward center of path curvature


E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt

bull Relations for tangential and normal accelerationalso apply for particle moving along a space curve

bull The plane containing tangential and normal unit

vectors is called the oscul atin g pl ane

nt b eee

bull The normal to the osculating plane is found from

binor mal e

nor mal pr inci pale

b

n

bull cceleration has no component along the binormal


E d i t i on

i nS I Uni t s

Sam$le Prolem ampampamp25678T$62



copy 2013 The McGraw-Hill Companies Inc All rights reserve )) 0)

motorist is traveling on a curvedsection of highway of radius mat the speed of 9 kmh Themotorist suddenly applies the brakescausing the automobile to slowdown at a constant rate Knowingthat after 8 s the speed has beenreduced to kmh determine theacceleration of the automobile

immediately after the brakes have been applied

5678T$62

bull Define your coordinate system

bull alculate the tangential velocity and

tangential acceleration

bull Determine overall acceleration magnitudeafter the brakes have been applied

bull alculate the normal acceleration


E d i t i on

i nS I Uni t s

Sam$le Prolem ampamp5678T$62 bull Define your coordinate system




))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +

km ) m ) h9 kmh - 9 ms

h ) km s kmh - ms

)) 0+

5678T$62 bull Define your coordinate system

et

en

bull Determine velocity and acceleration in

the tangential direction

bull The deceleration constant therefore

bull $mmediately after the brakes are applied

the speed is still ms

3 + 38+n t a a a


E d i t i on

i nS I Uni t s





In ampamp a race scheduled at the eas Motor Speedway wascancelled because the normal accelerations were too high and

caused some drivers to eperience ecessive gloads (similar to

fighter pilots) and possibly pass out What are some things that

could be done to solve this problem

Some possibilities

9educe the allowed speed

Increase the turn radius

(difficult and costly)

ave the racers wear gsuits

Vetor Mehanis $or Enineers Dna(isT

en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving5678T$62



copy 2013 The McGraw-Hill Companies Inc All rights reserve )) 0-

5678T$62




bull Determine overall acceleration

magnitude


The tangential acceleration of thecentrifuge cab is given by

where t is in seconds and at is inms $f the centrifuge starts from

fest determine the total acceleration

magnitude of the cab after )

seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D




$n the side view the tangentialdirection points into the 991260 page991261

Define your coordinate system

et

en

en

op 6iew

Determine the tangential velocity

t a t

t t

t v t dt t t

) mst v

Determine the normal acceleration

8) ms8

t

nvar

Determine the total acceleration magnitude

8) amp 3+3)+mag n t a a a

88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving




a+ The accelerations would remain the same

b+ The an would increase and the at would decrease

c+ The an and at would both increased+ The an would decrease and the at would increase

Notice that the normal

acceleration is much higher than

the tangential acceleration

What would happen if for agiven tangential velocity and

acceleration the arm radius was

doubled


en t h

E d i t i on

i

nS I Uni t s





5y knowing the distance to the aircraft and theangle of the radar air traffic controllers can

track aircraft

Fire truck ladders can rotate as well as etend

the motion of the end of the ladder can be

analy+ed using radial and transverse

components


en t h

E d i t i on

i

nS I Uni t s





bull The position of a particle P isexpressed as a distance r from the

origin to P amp this defines the

radial direction er The transverse

direction e983153 is perpendicular to er

v 983101 r er 983083 r 983153e

983153

bull The particle velocity vector is

bull The particle acceleration vector is

a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s


bull 0e can derive the velocity and acceleration




0e can derive the velocity and acceleration

relationships by recogniing that the unit vectors

change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r


bull 5imilarly the particle acceleration vector is

er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4




If you are travelling in a perfect

circle what is always true about

radial3transverse coordinates andnormal3tangential coordinates

a+ The er direction is identical to the en direction

b+ The e983153 direction is perpendicular to the en directionc+ The e983153 direction is parallel to the er direction


en t h

E d i t i on

i n

S I Uni t s


bull 0hen particle position is given in cylindrical




p p g y

coordinates it is convenient to express the

velocity and acceleration vectors using the unit

vectors and k ee

bull Bosition vector

k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s




copy 2013 The McGraw-Hill Companies Inc All rights reserve )) 1+

Rotation of the arm about 6 is defined

by 983153 = )t where 983153 is in radians and t

in seconds ollar 4 slides along the

arm such that r - 9 )t where r is

in meters

fter the arm has rotated through o

determine ( a) the total velocity of the

collar (b ) the total acceleration of the

collar and ( c ) the relative acceleration

of the collar with respect to the arm

5678T$62bull Dvaluate time t for 983153 - o

bull Dvaluate radial and angular positions

and first and second derivatives attime t

bull alculate velocity and acceleration in

cylindrical coordinates

bull Dvaluate acceleration with respect to

arm


en t h

E d i t i on

i n

S I Uni t s

Sam$le Prolem ampampamp(5678T$62




bull Dvaluate time t for 983153 - o

s89)rad

)

t

t

bull Dvaluate radial and angular positions and first

and second derivatives at time t

sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s

Sam$le Prolem ampampamp(bull alculate velocity and acceleration




r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s


bull Dvaluate acceleration with respect to arm




otion of collar with respect to arm is rectilinear

and defined by coordinate r

sm r a A B


en t h

E d i t i on

i n

S I Uni t s






bull alculate the angular velocity after

three revolutions

bull D

etermine overall accelerationmagnitude

bull alculate the radial and transverse

accelerations

The angular acceleration of thecentrifuge arm varies according to

where 983153 is measured in radians $f the

centrifuge starts from rest determine the

acceleration magnitude after the gondola

has travelled two full rotations

983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di




$n the side view the transversedirection points into the 991260 page991261

efine your coordinate system

e983153

er

er

op 6iew

Determine the angular velocity

valuate the integral

983153 983101 983153 3rads+

983153d 983153 983101 983153d 983153cceleration is a function

of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er


Determine the angular acceleration

983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads

Find the radial and transverse accelerationsa 983101 r 983085 r 983153 983080 983081er

983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +

3 )+ amp mag r a a a

msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r




You could now have additional acceleration terms This might

give you more control over how quickly the acceleration of the

gondola changes 3this is known as the ltonset rate+

What would happen if youdesigned the centrifuge so

that the arm could etend

from 2 to amp meters

a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

i n

S I Uni t s

Contents

)) +

Introduction

Rectilinear Motion Position

elocit Acceleration

etermination of the Motion of aParticle



niform Rectilinear+Motion

niforml Accelerated Rectilinear+

Motion


Relative Motion



e$endent Motion


ra$hical Solution of Rectilinear+

Motion Prolems

ther ra$hical MethodsCurvilinear Motion Position elocit

Acceleration


Rectangular Com$onents of elocit

and Acceleration

Motion Relative to a 0rame in

TranslationTangential and 1ormal Com$onents


Sam$le Prolem ampampamp2






E d i t i on

i n

S I Uni t s

))









E d i t i on

i n

S I Uni t s

Introduction

)) -








Flift

Fdrag




E d i t i on

i n

S I Uni t s

Introduction

))






dimensions




E d i t i on

i n

S I Uni t s


))










function eg t t x





E d i t i on

i n

S I Uni t s


)) 0





t

xv

t

x

t

lim

Averag e velocity



dt dx

t xv

t

lim

eg

)

t t dt dxv

t t x

T





E d i t i on

in

S I Uni t s


)) 1



va

t

lim

t dt dva

t t v

dt xd

dt dv

t va

t

)

)eg

lim







T i





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2





velocity must ero



T i





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

i n

S I Uni t s

))









E d i t i on

i n

S I Uni t s

Introduction

)) -








Flift

Fdrag




E d i t i on

i n

S I Uni t s

Introduction

))






dimensions




E d i t i on

i n

S I Uni t s


))










function eg t t x





E d i t i on

i n

S I Uni t s


)) 0





t

xv

t

x

t

lim

Averag e velocity



dt dx

t xv

t

lim

eg

)

t t dt dxv

t t x

T





E d i t i on

in

S I Uni t s


)) 1



va

t

lim

t dt dva

t t v

dt xd

dt dv

t va

t

)

)eg

lim







T i





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2





velocity must ero



T i





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

i n

S I Uni t s

Introduction

)) -








Flift

Fdrag




E d i t i on

i n

S I Uni t s

Introduction

))






dimensions




E d i t i on

i n

S I Uni t s


))










function eg t t x





E d i t i on

i n

S I Uni t s


)) 0





t

xv

t

x

t

lim

Averag e velocity



dt dx

t xv

t

lim

eg

)

t t dt dxv

t t x

T





E d i t i on

in

S I Uni t s


)) 1



va

t

lim

t dt dva

t t v

dt xd

dt dv

t va

t

)

)eg

lim







T i





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2





velocity must ero



T i





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

i n

S I Uni t s

Introduction

))






dimensions




E d i t i on

i n

S I Uni t s


))










function eg t t x





E d i t i on

i n

S I Uni t s


)) 0





t

xv

t

x

t

lim

Averag e velocity



dt dx

t xv

t

lim

eg

)

t t dt dxv

t t x

T





E d i t i on

in

S I Uni t s


)) 1



va

t

lim

t dt dva

t t v

dt xd

dt dv

t va

t

)

)eg

lim







T i





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2





velocity must ero



T i





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

i n

S I Uni t s


))










function eg t t x





E d i t i on

i n

S I Uni t s


)) 0





t

xv

t

x

t

lim

Averag e velocity



dt dx

t xv

t

lim

eg

)

t t dt dxv

t t x

T





E d i t i on

in

S I Uni t s


)) 1



va

t

lim

t dt dva

t t v

dt xd

dt dv

t va

t

)

)eg

lim







T i





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2





velocity must ero



T i





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

i n

S I Uni t s


)) 0





t

xv

t

x

t

lim

Averag e velocity



dt dx

t xv

t

lim

eg

)

t t dt dxv

t t x

T





E d i t i on

in

S I Uni t s


)) 1



va

t

lim

t dt dva

t t v

dt xd

dt dv

t va

t

)

)eg

lim







T i





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2





velocity must ero



T i





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

in

S I Uni t s


)) 1



va

t

lim

t dt dva

t t v

dt xd

dt dv

t va

t

)

)eg

lim







T i





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2





velocity must ero



T i





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

in

S I Uni t s

Conce$t 3ui4

)) 2





velocity must ero



T i





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

n

S I Uni t s


)) )3


t t x

) t t dt dxv

t

dt

xd

dt

dva )

at t - s x - ) m v - vma x - ) ms a -











E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

n

S I Uni t s


)) ))








a spring drag






E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

n

S I Uni t s


)) )+

a a t

v t

v

dv a t dt 3 +dv

a t

dt

v dv a x dx

v x

v xv dv a x dx

a a x

anddx dv

dt av dt

dv

v a vdx

v t

v

dv dt a v

x v

x v

v dvdx a v

a a v

3 +dv a vdt



i n





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

n

S I Uni t s


)) )

Determine






5678T$62








i n





E d i t i on

n

S I Uni t s


)) )-

t vt vdt dv

a

dt

dv

t t v

v

8)98)9

sm8)9

t t v

s

m8)9sm)

)

8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

dy

t t y

y

s

m9s

m)m t t t y



i n





E d i t i on

S I Uni t s


)) )


s

m8)9

s

m)

t t v

s)9)t


s)9)s

m9s)9)

s

m)m

s

m9

s

m)m

y

t t t y

m) y


i n





E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

s8

smeaningles s)

t

t

s8

s

m8

)9s

m

)s8

s

m8)9

s

m)

v

t t v

s

mv


i n





E d i t i on

S I Uni t s


)) )0







to their velocity


kva

5678T$62




v3 x+


i n





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





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i n

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x

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t t t eee

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sin


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fter substituting

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average ms8 st t

v

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en t h

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nS I Uni t s





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magnitude






seconds

3ms +t a t


en t h

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nS I Uni t s

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et

en

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op 6iew


t a t

t t

t v t dt t t

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en t h

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i

nS I Uni t s

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doubled


en t h

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nS I Uni t s






track aircraft




components


en t h

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i

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v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

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S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

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i n

S I Uni t s

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en t h

E d i t i on

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S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

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S I Uni t s









in meters












arm


en t h

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i n

S I Uni t s






s89)rad

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t

t



sm

sm9m8))9

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srad)

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t


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r

r

v

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m9

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a

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en t h

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bull D



accelerations






983153 983101 983153 3rads+


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i n

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e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

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3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

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i n

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D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

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msmag a

Magnitude


en t h

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r










a 983101 r 983085 r 983153

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E d i t i on

n

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a

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v

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8)9)8)9)

8)9)

t t yt ydt t dy

t vdt

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y

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m9s

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i n





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m8)9

s

m)

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s

m)m

s

m9

s

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y

t t t y

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m9

s

m)m

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smeaningles s)

t

t

s8

s

m8

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m

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s

m8)9

s

m)

v

t t v

s

mv


i n





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kva

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v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


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kt

t x t kt kt

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dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

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)) +)






bottom of the lake



than amp

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i n

S





E d i t i on

SI Uni t s

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)) ++

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position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


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v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

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))ev

8 msv



acceleration is ero

This occurs when

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max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

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i n

S I U




h

E d i t i on

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with respect to A

A B A B

vvv




i n

S I U




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)) )





















elevator


i n

S I U




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Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

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B

B



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v

E E

E

s

mm

s

m


i n

S I U




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smeaningles s9

t

t



E y

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8)9)

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t v E B

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i n

S I U




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or

or

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aaadt

dv

dt

dv

dt

dv

vvvdt

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dt

dx

dt

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i n

S I Un




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E

i n

S I Un




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5 6

mm mm ) s

s s

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7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

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i n

S I Un




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x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


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i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


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i n

S I Uni





di t i on

ts


)) -+






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i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


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i n

S I Uni t s





i on

s


)) -2

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t

v dva

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r r

r



v



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u

u P uu P

u

P

du

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uu

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d

du

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du

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du

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du

d f

du

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du

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du

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du

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du

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du

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k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

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k a aia

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dt

yd i

dt

xd a

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i n

S I Uni t s






)) +



$ a g ya xa $ y x


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)

$ g t yv yt v x

v g t vvvv

y x

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SOLUION

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Conce$t 3ui4




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))




a+ launch angle of





E d i t i on

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bull alculate v0=0


E d i t i on

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n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

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t

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E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


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bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

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5678T$62



t t - s


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5678T$62



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x

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en



v4 vt et


radius of curvature

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en t h

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magnitude






seconds

3ms +t a t


en t h

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t a t

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doubled


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track aircraft




components


en t h

E d i t i on

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v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

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edt

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dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

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d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

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en t h

E d i t i on

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S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

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en t h

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in meters












arm


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t



sm

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srad)

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t


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r

r

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accelerations






983153 983101 983153 3rads+


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3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

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983153

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983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

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msmag a

Magnitude


en t h

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r










a 983101 r 983085 r 983153

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kva

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v t dv dv

a kv k dt kt dt v v

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kt

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k

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and

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S





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bottom of the lake


v t

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v x

v xv dv a x dx

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x v

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(b)

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i n

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i n

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A B A B

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i n

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i n

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s

m9

s

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983101 x D983080 983081

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x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

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mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

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E

i n

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A D B

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x x x

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v

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mm 3+

s

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B

a a a

a

mm mm

s s Ba 7 + 7


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i n

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di t i on

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velocity of block B

Solution steps


coordinate system



get velocity


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i n

S I Un




di t i on

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x

y4





get velocity



ms amp Bv

ms B

v



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di t i on

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2

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2

2

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amp22

amp(2

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v

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v

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BC

AB

dx

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tan



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i n

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ti on

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curvilinear motion




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i n

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ti on

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r


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i n

S I Uni t s





ti on

s


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t

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3vector+

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t

r d r v

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r



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u

u P uu P

u

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k $ yi xr


k v viv

k $ yi xk dt

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k a aia

k $ yi xk dt

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5678T$62







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)

3+ f y y t g t

) m 398) ms +

t

)

g t

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the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


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i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



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r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

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53

53

j i v

v j i

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) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

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a+

b+

c+

d+o

o


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i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



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i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


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i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

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dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

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nt b eee


binor mal e


b

n



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i nS I Uni t s







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i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









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en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


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i n

S I Uni t s





r

r

r

v

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r v

sr v

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smsrad)m8)

m9

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a

aaaa

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sm9)

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i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



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of position so use

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3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

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E d i t i on

S I Uni t s


)) )


s

m9

s

m)m

t t t y

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smeaningles s)

t

t

s8

s

m8

)9s

m

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s

m8)9

s

m)

v

t t v

s

mv


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to their velocity


kva

5678T$62




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v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

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components


en t h

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v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

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p p g y



vectors and k ee


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en t h

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r 983083 38+38+ 983083 983080 983081e983153

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r 983083

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Magnitude


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r










a 983101 r 983085 r 983153

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kt

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kt

ek

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S





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v

kxvv

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S





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v

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(b)

(c)

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dvv a v

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v dvdx

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S





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x v

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acceleration is ero

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i n

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motion




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mm 3+

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velocity of block B

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coordinate system



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i n

S I Un




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x

y4





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ms amp Bv

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2

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)))

v

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du

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k $ yi xr


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dt

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bull alculate v0=0


E d i t i on

i nS I Uni t s

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n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

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r j i

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v j i

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5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


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i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

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dv

dt

ed ve

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dv

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butv

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dsdsd e

d

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fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

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nt b eee


binor mal e


b

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i nS I Uni t s







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8 mstan

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t

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aa 7 7

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average ms8 st t

v

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h ) km s kmh - ms

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et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









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en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

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sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

SI Uni t s


)) )1

5678T$62


ln

v t

v

v t dv dv

a kv k dt kt dt v v

kt evt v


)

kt

t x t kt kt

dxv t v e

dt

dx v e dt x t v ek

kt

ek

v

t x

)


i n

S





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





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IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




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v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

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)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

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9)8) t t t y E B

s

smeaningles s9

t

t



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s

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E

i n

S I U




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or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

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))





















E

i n

S I Un




E d i t i on

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))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

SI Uni t s


)) )2


kxvv

dxk dvdxk dvkvdx

dvva

xv

v

kxvv

bull lternatively

)v

t v

k

vt x

kxvv

or v

t veevt v

kt kt

kt ek

vt x )with

and

then


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




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Uni t s


)) )





















elevator


i n

S I U




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Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




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))


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s

smeaningles s9

t

t



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m) E y

8)9)

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s

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i n

S I U




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or

or

C B AC B A

C B AC B A

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dv

dt

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i n

S I Un




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E

i n

S I Un




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ni t s


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5 6

mm mm ) s

s s

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7 7

A A Av v a t

t t

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A A

v v a x x

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i n

S I Un




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x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

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di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


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during a round of

boing


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i n

S I Uni





di t i on

ts


)) -+






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i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

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)

v

v



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t t a-t t v x x

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curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

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lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

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)) 3

u P


u

u P uu P

u

P

du

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uu

limlim


du

d

du

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du

P d

du

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du

d f

du

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du

d

P du

P d

du

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du

d P

du

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du

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i n

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k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

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k a aia

k $ yi xk dt

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dt

yd i

dt

xd a

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i n

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)) +



$ a g ya xa $ y x


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$ g t yv yt v x

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distance



E d i t i o

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n Conce$t 3ui4

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a+ launch angle of





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5678T$62





) m

bull alculate v0=0


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i nS I Uni t s

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n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

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the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


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i nS I Uni t s





n

)) 2




making a pass




make a safe landing


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n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

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5678T$62






acceleration




E d i t i on

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3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

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5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

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53

53

j i v

v j i

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) 3ms +

5 3

5 3

j i a

a j



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i nS I Uni t s

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a+

b+

c+

d+o

o


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i nS I Uni t s









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i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



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i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

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nt b eee


binor mal e


b

n



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8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

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i

nS I Uni t s

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doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

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i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

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srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

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i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +3






bottom of the lake


v t

vdv a t dt

v x

v xv dv a x dx

v t

v

dvdt

a v

x v

x v

v dv

dx a v

(a)

(b)

(c)

(d)

$y


i n

S





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

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))ms7a )a 7 o

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ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

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m9

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r r

r

a

aaaa

r r a

r r a

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sm9)

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en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

SI Uni t s

Conce$t 3uestion

)) +)






bottom of the lake



than amp

$y


i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

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with respect to A

A B A B

vvv




i n

S I U




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)) )





















elevator


i n

S I U




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Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

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B



t t v y y

v

E E

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s

mm

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m


i n

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s

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t

t



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i n

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or

or

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C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

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dt

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dt

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i n

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E

i n

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5 6

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s s

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7 7

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v v a x x

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x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

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)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

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n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

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n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



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i nS I Uni t s







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E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) ++

5678T$62




position1




after onehalf lap



)a v






i n

S





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

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))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




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)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


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))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

SI Uni t s

rou$ Prolem Solving

)) +

dvv a v

dx

x v

x v

v dvdx

a v


ltiven )a v

vo - r - m

(ind v after = lap

aximum speed





)3+ 8 m x r


i n

S





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

IUni t s

rou$ Prolem Solving

)) +-

x v

x v

v dvdx a v


8

)

v

vdx dvv


)8 ln )

v

v

83 + ln ) ln )3+v

ln ) ) )98- )8v

)8

)v e



i n

S I





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




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Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

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s

smeaningles s9

t

t



E y

m) E y

8)9)

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s

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i n

S I U




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or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


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i n

S I Un




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ni t s


))





















E

i n

S I Un




E d i t i on

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))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


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i n

S I Un




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)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

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mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


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during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

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)

v

v



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t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

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lim

t

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t dt

r r

r


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i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



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i n

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)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

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du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

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du

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i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


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$ g t yv yt v x

v g t vvvv

y x

$ y y x x






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i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

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E d i t i o

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))

SOLUION

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Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





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n $ g

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5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

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n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


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i nS I Uni t s





n

)) 2




making a pass




make a safe landing


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i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

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5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

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53

53

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v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


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i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



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i nS I Uni t s







5678T$62







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i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

IUni t s

rou$ Prolem Solving

)) +

)8 )v e Solve for v

)8

))ev

8 msv



acceleration is ero

This occurs when

) v

)maxv

max msv


i n

S I





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

Uni t s


)) +





vt x xvt x x

dt vdx

vdt

dx

t x

x

constant






motion




i n

S I





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





E d i t i on

Uni t s


)) +0





motion




i n

S I




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

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a+

b+

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o


E d i t i on

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E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

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v tv e



E d i t i on

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t t ee and

t t t eee

d

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e

e

t n

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t

sinlimlim

sin


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t evv



dt

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ds

d

d

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dt

dv

dt

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dt

dv

dt

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d

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fter substituting

vadt dvaeve

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E d i t i on

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dvae

ve

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binor mal e


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7 7 7

ms ms

average ms8 st t

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+

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h ) km s kmh - ms

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et

en






3 + 38+n t a a a


E d i t i on

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en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

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i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

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S I Uni t s









in meters












arm


en t h

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S I Uni t s






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t

t



sm

sm9m8))9

r

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srad)

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t


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r

r

r

v

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r v

sr v

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S I Uni t s








sm r a A B


en t h

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i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

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e983153

er

er

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of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

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en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

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er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

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msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




h

E d i t i on

Uni t s


)) +1




constantv t

v


)

x t

x


constant

v x

v x




i n

S I U




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

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with respect to A

A B A B

vvv




i n

S I U




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)) )





















elevator


i n

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Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

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))


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t

t



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i n

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ni t s


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or

or

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C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

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i n

S I Un




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E

i n

S I Un




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5 6

mm mm ) s

s s

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7 7

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t t

mm mm mm

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x D

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t

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983270

983272983271 983286

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mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

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d i t i on

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constant

mm mm

s s

A D B

A D B

B

x x x

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v

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mm 3+

s

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B

a a a

a

mm mm

s s Ba 7 + 7


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i n

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di t i on

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velocity of block B

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coordinate system



get velocity


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i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



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)) -)




2

(2

2

2

2

amp22

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v

dvt t t v

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)

)))

v

v

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)

v

v



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i n

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s


)) -


BC

AB

dx

dv

va

tan



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i n

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ti on

s


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curvilinear motion




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i n

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ti on

s


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r


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ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

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lim

t

r d r v

t dt

r r

r


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i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

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)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

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du

P d

du

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du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

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du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






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i nS I Uni t s






))







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distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

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))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

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n Conce$t 3ui4

))




a+ launch angle of





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)) 0





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bull alculate v0=0


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i nS I Uni t s

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n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




h

E d i t i on

Uni t s


)) +2



i n

S I U




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




h

E d i t i on

Uni t s


)) 3







with respect to A

A B A B

vvv




i n

S I U




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

Uni t s


)) )





















elevator


i n

S I U




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

Uni t s


)) +




)

s

m9

s

m)8m)

s

m8)9

s

m)8

t t at t v y y

t at vv

B

B



t t v y y

v

E E

E

s

mm

s

m


i n

S I U




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

Uni t s


))


9)8) t t t y E B

s

smeaningles s9

t

t



E y

m) E y

8)9)

8)9)8

t v E B

s

m8))9

E Bv


E

i n

S I U




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

ni t s


)) -









or

or

C B AC B A

C B AC B A

aaadt

dv

dt

dv

dt

dv

vvvdt

dx

dt

dx

dt

dx


E

i n

S I Un




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

ni t s


))





















E

i n

S I Un




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

ni t s


))




5 6

mm mm ) s

s s

7 8

7 7

A A Av v a t

t t

mm mm mm

s s

A A A A A

A A

v v a x x

a a


E

i n

S I Un




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




E d i t i on

ni t s


)) 0


x D

983101 x D983080 983081

983083 v D

t

x D 983085 x D983080 983081 983101 mms

983270

983272983271 983286

983288983287 )s983080 983081983101) mm





mm ) mm

A D B A D B

A A D D B B

B B

x x x x x x

x x x x x x

x x

5 6 mm B B x x+ 7 +


E

i n

S I Un




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




d i t i on

ni t s


)) 1


constant

mm mm

s s

A D B

A D B

B

x x x

v v v

v

mm mm

s s Bv 7 + 7

mm 3+

s

A D B

B

a a a

a

mm mm

s s Ba 7 + 7


E d

i n

S I Un




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




di t i on

i t s

rou$ Prolem Solving

)) 2



velocity of block B

Solution steps


coordinate system



get velocity


E d

i n

S I Un




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

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i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




di t i on

it s

rou$ Prolem Solving

)) -3


x

y4





get velocity



ms amp Bv

ms B

v



E d

i n

S I Uni




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

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dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




di t i on

it s


)) -)




2

(2

2

2

2

amp22

amp(2

amp2

amp2

amp2


Ti(e 6s7


during a round of

boing


E d

i n

S I Uni





di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




di t i on

ts


)) -+






E d

i n

S I Uni t





di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




di t i on

ts


)) -






E d

i n

S I Uni t

th hi l M th d




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




i t i on

ts


)) --


)

))

) curve underarea

v

v

dvt t t v

t v x x

using dv - a dt

)

)))

v

v

dt at t t v x x

)

)

v

v



C t

t t a-t t v x x

centroidof abscissa

curveunderarea )))


E d i

i n

S I Uni t s

th hi l M th d




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




i t i on

s


)) -


BC

AB

dx

dv

va

tan



E d i

i n

S I Uni t s





ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




ti on

s


)) -


curvilinear motion




E d i t

i n

S I Uni t s





ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




ti on

s


)) -0




r


E d i t

i n

S I Uni t s





ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




ti on

s


)) -1

lim

t

s dsv

t dt


3vector+

$nstantaneous speed

3scalar+

lim

t

r d r v

t dt

r r

r


E d i t

i n

S I Uni t s





i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




i on

s


)) -2

lim

t

v dva

t dt

r r

r



v



E d i t i

i n

S I Uni t s






)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





)) 3

u P


u

u P uu P

u

P

du

P d

uu

limlim


du

d

du

P d

du

P d

du

P d f P

du

d f

du

P f d



du

d

P du

P d

du

P d

du

d P

du

P d

du

P d



E d i t i

i n

S I Uni t s






)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





)) )


k $ yi xr


k v viv

k $ yi xk dt

d$

dt

dyi

dt

dxv

$ y x


k a aia

k $ yi xk dt

$ d

dt

yd i

dt

xd a

$ y x


E d i t i o

i n

S I Uni t s






)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





)) +



$ a g ya xa $ y x


$ y x vvv $ y x


)

$ g t yv yt v x

v g t vvvv

y x

$ y y x x






E d i t i o

i nS I Uni t s






))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





))







5678T$62







distance



E d i t i o

i nS I Uni t s






)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





)) -

SOLUION







E d i t i o

i nS I Uni t s






))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





))

SOLUION

ori+ontal distance








E d i t i o

i nS I Uni t s

Conce$t 3ui4




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




n Conce$t 3ui4

))




a+ launch angle of





E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




n $ g

)) 0





5678T$62





) m

bull alculate v0=0


E d i t i on

i nS I Uni t s

rou$ Prolem Solving




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




n $ g

)) 1


the ydirection


(ind vo

)

3+ f y y t g t

) m 398) ms +

t

)

g t

)9 st


the direction

3 + x x v t v t

) m 3 +3)9 s+v

ms ) kmhv


E d i t i on

i nS I Uni t s





n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




n

)) 2




making a pass




make a safe landing


E d i t i on

i nS I Uni t s





n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




n

)) 3

bull D






A Br

A B A B r r r


A Bv


A Ba


A B A B aaa





E d i t i on

i nS I Uni t s












5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153











5678T$62






acceleration




E d i t i on

i nS I Uni t s







3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153






3v B+- a B- ) ms 3y A+ - m



t t - s


E d i t i on

i nS I Uni t s





5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

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sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




5678T$62



t t - s


E d i t i on

i nS I Uni t s





5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

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r j i

)

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53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

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7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

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r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

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sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




5678T$62



5 5 3

r r r

5 5 3 v v v 5 53a a a

3m+

53

53

j i r

r j i

)

) 3ms+

53

53

j i v

v j i

)

) 3ms +

5 3

5 3

j i a

a j



E d i t i on

i nS I Uni t s

Conce$t 3ui4







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153







a+

b+

c+

d+o

o


E d i t i on

i nS I Uni t s









E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153








E d i t i on

i nS I Uni t s






x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

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dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


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E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

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m9

)sm v

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r

a

aaaa

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r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

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sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





x

y

et

en



v4 vt et


radius of curvature

dv v

dt

t n

a e e

v tv e



E d i t i on

i nS I Uni t s










t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

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dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

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nt b eee


binor mal e


b

n



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i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

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ms

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3 ms+ 8 ms m

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7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

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i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

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dt

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dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

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i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

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srad

srad)

rad)

t

t


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i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

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smsrad)m8)

m9

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r

a

aaaa

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r r a

)

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sm9

srad)sm9sradm8)

sm9)

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sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153









t t ee and

t t t eee

d

ed e

ee

e

e

t n

nnt

t

sinlimlim

sin


E d i t i on

i nS I Uni t s





t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

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i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

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dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

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sr v

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m9

)sm v

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r

a

aaaa

r r a

r r a

)

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sm9

srad)sm9sradm8)

sm9)

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sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




t evv



dt

ds

ds

d

d

ed ve

dt

dv

dt

ed ve

dt

dv

dt

vd a t t

butv

dt

dsdsd e

d

ed n

t

fter substituting

vadt dvaeve

dt dva nt nt








E d i t i on

i nS I Uni t s





va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




va

dt

dvae

ve

dt

dva nt nt




nt b eee


binor mal e


b

n



E d i t i on

i nS I Uni t s







5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153






5678T$62







E d i t i on

i nS I Uni t s





))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




))ms7a )a 7 o

8 mstan

ms

n

t

a

aa 7 7

3 ms+ 8 ms m

nvar

7 7 7

ms ms

average ms8 st t

v

a a t

+

7 7 7 7 +


h ) km s kmh - ms

)) 0+


et

en






3 + 38+n t a a a


E d i t i on

i nS I Uni t s









Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153








Some possibilities






en t h

E d i t i on

i

nS I Uni t s





5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




5678T$62





magnitude






seconds

3ms +t a t


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving

D






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153






et

en

en

op 6iew


t a t

t t

t v t dt t t

) mst v


8) ms8

t

nvar



88 msmag a


en t h

E d i t i on

i

nS I Uni t s

rou$ Prolem Solving












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153












doubled


en t h

E d i t i on

i

nS I Uni t s






track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





track aircraft




components


en t h

E d i t i on

i

nS I Uni t s









v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153








v 983101 r er 983083 r 983153e

983153



a 983101 r 983085 r 983153 983080 983081er 983083 r 983153 983083r 983153983080 983081e

983153

r er r


en t h

E d i t i on

i n

S I Uni t s








change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153






change direction

r r e

d

ed e

d

ed

dt

d e

dt

d

d

ed

dt

ed r r

dt

d e

dt

d

d

ed

dt

ed r

er er

edt

d r e

dt

d r

dt

ed r e

dt

d r er

dt

d v

r

r r

r r



er r er r

dt

ed

dt

d r e

dt

d r e

dt

d

dt

d r

dt

ed

dt

d r e

dt

r d

edt

d r e

dt

d r

dt

d a

r

r r

r

r er r


en t h

E d i t i on

i n

S I Uni t s

Conce$t 3ui4










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153










en t h

E d i t i on

i n

S I Uni t s






p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




p p g y



vectors and k ee


k $ e r


k $ e e dt

r d v


k $ e e dt

vd a


en t h

E d i t i on

i n

S I Uni t s









in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153








in meters












arm


en t h

E d i t i on

i n

S I Uni t s






s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153





s89)rad

)

t

t



sm

sm9m8))9

r

t r t r

srad

srad)

rad)

t

t


en t h

E d i t i on

i n

S I Uni t s





r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




r

r

r

v

vvvv

r v

sr v

) tan

smsrad)m8)

m9

)sm v

r r

r

a

aaaa

r r a

r r a

)

tan

sm9

srad)sm9sradm8)

sm9)

srad)m8)sm

sm) a


en t h

E d i t i on

i n

S I Uni t s








sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153






sm r a A B


en t h

E d i t i on

i n

S I Uni t s







three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153






three revolutions

bull D



accelerations






983153 983101 983153 3rads+


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Dfi di






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153






e983153

er

er

op 6iew



983153 983101 983153 3rads+


of position so use

983153

3983152 +

983101 983153

983153

3+3983152 +

983282 983153

d 983153

983101

983153d 983153

983153

983282983153

983101 3983152 +983273983275 983289983291


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

D t i th l l it




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153




er



983153 983101 899 rads

983153 983101 3983152 +983273983275 983289983291

983153 983101 983153 - 3+3983152 + 983101 8 rads


983083 r 983153 983083 r 983153983080 983081e983153

983101 983085 38+3899+983080 983081e

r 983083 38+38+ 983083 983080 983081e983153

983101 983085) e

r 983083

e983153 3ms +


msmag a

Magnitude


en t h

E d i t i on

i n

S I Uni t s

rou$ Prolem Solving

Wh t ld h if

r










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153










a 983101 r 983085 r 983153

983080 983081er 983083 r 983153 983083r 983153983080 983081e983153

11 Lecture Ppt

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Transcript of 11 Lecture Ppt