Mechanics 105

Motion diagrams, position-time graphs, etc.

Average and instantaneous velocity

Acceleration

Particle under constant acceleration

Freefall

Motion in one dimension (chapter two)

Mechanics 105

Motion diagrams

Motion diagram:

Mechanics 105

Position vs. time

x vs. t

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 35

time (seconds)

po

sit

ion

(m

)

Mechanics 105

Velocity vs. time

0

2

4

6

8

10

0 5 10 15 20 25 30

time (seconds)

po

siti

on

(m

)

-1.5

-1

-0.5

0

0.5

1

0 5 10 15 20 25 30

time (seconds)

velo

city

(m

/s)

Mechanics 105

Position vs. time

Demo - ~constant velocity

Conceptests

http://webphysics.davidson.edu/physletprob/ch7_in_class/in_class7_1/mechanics7_1_1.html

http://webphysics.davidson.edu/physletprob/ch8_problems/ch8_1_kinematics/default.html

Mechanics 105

only depends on beginning and end points and time interval vector (scalar in 1-d only) total displacement = integral of v(t)dt

examples:

A person runs 1 km in 5 minutes, then walks another 2 km in 20 minutes. What is their average velocity over the entire 3 km?

Using the velocity vs. time graph shown earlier, find the average velocity and the total displacement

Average velocity

f

i

f

i

x

x

x

x

if dtdt

tdxdttvxxx

)()(

Mechanics 105

Instantaneous velocity

limit of average velocity as interval goes to zero tangent to x(t) - derivative of x(t) vs. t examples

A ball rolling down a slope has a position described by the equation

What is the equation describing the instantaneous velocity? ConcepTests

22 )/5.0()/1.0(50.0)( tsmtsmmtx

Mechanics 105

Particle under constant velocity

relation between initial and final displacement comes from the definition of average velocity

average equals instantaneous

tvxxvt

xif

Mechanics 105

Acceleration

average

instantaneous

t

vva if

2

2

0lim)(

dt

xd

dt

dv

t

vta

t

Mechanics 105

Particle moving under constant acceleration

tavvt

vva if

if

Displacement of particle = integral of velocity as a function of time

From definition of average acceleration:

t)(tattvxx

tatvdtatvdttvx

iif

t

t

ii

t

t

f

i

f

i

2

1

2

1)()(

2

2

Mechanics 105

Particle moving under constant acceleration

tvvxtt

vvtvxattvxx

t

vvaatvv

ifiif

iiiif

ifif

)(2

1)(

2

1

2

1

)(

22

Or (removing t)

Combining previous two equations (removing a)

)(2

)(

2

1)(

2

1

)(

22

2

22

ifif

ififiiiif

ifif

xxavv

a

vva

a

vvvxattvxx

a

vvtatvv

Mechanics 105

Particle moving under constant acceleration

Wide range of applications

Zero acceleration:

Free fall:

acceleration due to gravity = 9.80 m/s2 downward (be careful about sign)

If we define up to be the direction of the positive y axis, the equations of motion for a particle in free fall are:

tvxx iif

gtvvgttvyy iii and 2

1 2

Note that the velocity can be positive or negative

Mechanics 105

Particle moving under constant acceleration

ConcepTests

Demo – cart

More physlets

http://webphysics.davidson.edu/physletprob/ch8_problems/ch8_1_kinematics/default.html

Examples

Mechanics 105

Example - braking distance (from Giancoli, 2-10)

Estimate minimum stopping distance for a car traveling at 60 mph

1) Maximum (negative) acceleration? 5~8 m/s2 (dry road, good tires)

2) Typcial human response time? 0.3 ~ 1.0 sec (sober)

part 1 – distance before brakes applied:

part 2 – distance until car stops:

smmile

mhour

hour

miles/8.26

1

1610

sec3600

160

msm

smsmsm

a

vvxx 8.97

)/00.8(

)/8.26()/000.0(/04.8

2 2

2220

2

0

msmtvx 04.8sec)300.0)(/8.26(0

Mechanics 105

Example – Air bags (Giancoli 2-11)

If, instead of braking, the car in the previous example hit a tree, estiamte how fast the air bags need to inflate to do any good.

estimate a stopping distance ~ 1.00m

initial velocity = 26.8 m/s

final velocity = 0.00 m/s

first solve for a:

then find t

222

/35900.2

)/8.26(

2sm

m

sm

x

va f

sec0747.0/359

)/8.26/00.0()(2

sm

smsm

a

vvt if

Mechanics 105

Problem solving

Choice of coordinate system can simplify problem

Be consistent with signs (direction of chosen axis)

Often problems involve two or more objects with some common variable (time, final displacement, etc.)