File 1a Kinematika 1

8/17/2019 File 1a Kinematika 1

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Chapter 3Chapter 3

KINEMATICS IKINEMATICS IRahmat Rasyid M.SiRahmat Rasyid M.Si


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IntroductionIntroduction

The mechanics science is the sciencestudying about motion, force concept, andenergy.

They divided to the two form :• Kinematics

• Dynamics

Kinematics is the science studying the

motion without oo!ing the cause of motion. Dynamics is the science studying the force

cause of motion.


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Motion ?Motion ?

The ob"ect can referred motion if changeof the position and need time to do that.

Three variabe of the motion : Speed #eocity

$cceeration


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SpeedSpeed

The measurement to how fast the ob"ectmove.


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VelocityVelocity

%&pressing vaue and direction of speed orrate of change of position.

They divided to the two form :

• 'nstantaneous veocity

• $verage veocity


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The Difference Speed & VelocityThe Difference Speed & Velocity

Comparing both of the pictures

Aboe ! the car hae re"ain #peed

but the direction chan$e for eery

"o"ent% o' 'ith the picture

be#ide?


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Acceleration Acceleration


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Acceleration Acceleration

The e&pression that how fast the veocity ofchange.

They divided to the two form : 'nstantaneous acceeration

$verage acceeration


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Acceleration for Inclined (lane Acceleration for Inclined (lane

%ver greater theob"ect ange, henceever greater the

acceeration of ob"ect.

(ase :• )ow much vaue of the

acceeration if theob"ect in perpendicuarpane as shown picturebeside*


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)ree )all Motion)ree )all Motion

The ob"ect can freefa as conse+uenceof the earth

gravitation withacceeration ag-. m/s0


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Motion in * di"en#ionMotion in * di"en#ion

'n 12D, we usuay write position as x(t 1 ).

Since it3s in 12D, a we need to indicate directionis 4 or −.Dispacement in a time ∆t = t 2 - t 1 is

∆ x = x(t 2 ) - x(t 1 ) = x 2 - x 1

t

x

t 1 t 2

∆ x

∆ t

x 1

x 2 #o"e particle+# tra,ectory

in *-D


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*-D .ine"atic#*-D .ine"atic#

#eocity v is the 5rate of change of position6

$verage veocity V av in the time ∆ t = t 2 - t 1 is:

t

x

t t

)t ( x )t ( x v

12

12 av

∆

∆=

−

−≡

t

x

t 1 t 2

∆ x

x 1

x 2 tra,ectory

∆ t

V av = #lope of line connectin$ x 1 and x 2 %


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*-D .ine"atic#%%%*-D .ine"atic#%%%

(onsider imit t1 t0 'nstantaneous veocity v is defined as:

dt

t dxt v

)()( =

t

x

t 1 t 2

∆ x

x 1

x 2

∆ t

#o v(t 2 ) = #lope of line tan$ent to path at t 2 %

( )t

t xt v

t

∆

∆=

→∆

)(lim

0


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E/a"pleE/a"ple

7ind instantaneous veocity for every times, by7ind instantaneous veocity for every times, bye+uatione+uation x(t) = 5t 2

Answer : Answer : 7or the ne&t time7or the ne&t time , , t = t + ∆t , , so that:so that:

x(t +∆t) = 5(t + ∆t)2

then :then : ∆x = x(t +∆t) - x(t)

7inay :7inay :

∆t→0 then :then : vavg = 10t

( )t t t

t t t

t

xvavg ∆+=∆

∆+∆

=∆

∆

= 5105.10 2


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*-D .ine"atic#%%%*-D .ine"atic#%%%

• $cceeration a is the 5rate of change of veocity6

• $verage acceeration aav in the time ∆t = t 2 - t 1

is:

t

v

t t

)t ( v )t ( v

a 12

12

av ∆

∆

=−

−

≡

$nd then the

instantaneous acceeration a isdefined as:

2

2

dt

)t ( x d

dt

)t ( dv

)t ( a ==

dt

)t ( dx )t ( v =u#in$


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0ecap0ecap

'f the position x is !nown as a function of time,then we can find both veocity v and acceerationa as a function of time8

a dv dt

d x dt

= =

2

2

v dx

dt =

x x t = ( )

x

a

v t

t

t


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*-D Motion 'ith con#tant*-D Motion 'ith con#tant

accelerationacceleration 9y cacuus:

$so reca that

Since a is constant, we can integrate this usingthe above rue to find:

Simiary, since we can integrate againto get:

const t 1n

1dt t 1nn ++

= +∫

a dv

dt =

v dx

dt =

∫ ∫ +=== 0 v at dt adt av

0 0 2

0 x t v at 2

1dt )v at ( dt v x ++∫ ∫ =+==


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0ecap0ecap

at v v 0 +=

2 0 0 at 2

1t v x x ++=

a const =

x

a

v t

t

t


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Deriation!Deriation!

ugging in for t:

at v v 0 += 2

0 0 at 2

1t v x x ++=

Soving for t:

a

v v t 0

−=

2

0 0 0 0

a

v v a

2

1

a

v v v x x

−+

−+=

) x x ( a2 v v 0 2

0 2 −=−


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(roble" !(roble" !

%&press your comment for under the picture by%&press your comment for under the picture by

using average and instantaneous veocity andusing average and instantaneous veocity and

acceeration in point a, b c, d, e, f and g :acceeration in point a, b c, d, e, f and g :

t ;sec<

= ; m

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