Chapter 2: Kinematics

• 2.1 Uniform Motion

• 2.2 Instantaneous Velocity

• 2.3 Finding Position from Velocity

• 2.4 Motion with Constant Acceleration

• 2.5 Free Fall

• 2.6 Motion on an Inclined Plane

• 2.7* Instantaneous Acceleration

Stop to think 2.1 P38Stop to think 2.2 P44Stop to think 2.3 P48Stop to think 2.4 P54Stop to think 2.5 P61

Example 2.3 P 40Example 2.4 P 41Example 2.7 P 45Example 2.10 P 47Example 2.14 P 53Example 2.16 P 56Example 2.18 P 58

Motion in one dimension• Determining the signs of position, velocity and acceleration

1 s 2 s 3 s 4 s

xOrigin (x=0)

10 cm

20 cm

40 cm

70 cm

Motion along a straight line

Can be illustrated by position-versus-time graph:

x

t(s)1 2 3 4

Continuous (smooth) curve

Position vs time graphs

Interpreting a position graph1.What is the position at t =0min2.What is the position at t =30min3.What is the velocity at t = 20min4.What is the velocity at t = 50min5. What is the acceleration at t=20min6. If this is V vs. t graph, and x = 0 kmat t = 0min. What is the position at t = 80 min

Finding velocity from position graphically

Uniform Motion

• V(avg)= comstant• The position-vs-graph

is a straight line• Vs = ∆s/ ∆t

• Sf = Si + Vs ∆t

Instantaneous velocity

• Using motion diagrams and graphs

dt

ds

t

sV

ts

0

lim

Stop to think 2.2Which velocity-versus-time goes with the position –versus-time graph

C

Relating a velocity graph to a position graph

TThe value of the velocity atAny time equals the slope ofThe position graph

• Using calculus to find the velocity

Ex. A particle’s position is given by the function

1.What is particle’s position at t = 2s?

x = -8+6 = -2 m

2. What is the velocity at t = 2s

V|t=2 = -3(2)2+3=-9 m/s

mttx )3( 3

2( 3 3) /dx

V t m sdt

Finding position from Velocity

tf

ti

sif dtVSS

tfand between ti V curve volocity under the area s if SS

Example 2.9

1.Where is particle’s turning point?2.At what time does the particle reach the origin?

Motion with constant acceleration

t

VV

t

Va

if

taVV if

2)(2/12

)(

2

)(tatVt

VtaVt

VVs i

iiif

saViVa

VVVVt

VVs f

ififif

2

)(

2

)(

2

)( 22

Definition of acceleration

See page 57

If set t0=0s, ∆t = t

Example 2.13A rocket sled accelerates at 50m/s2 for 5.0 s. Coasts for 3.0 s, then deploys a parachute and decelerates at 3.0m/s2 until coming to a halt.

What is the maximum velocity of the rocket sled?What is the total distance traveled?

The apple and feather in this photograph are falling in a vacuum

Two objects dropped from the same height will, if air resistance can be neglectedHit the ground at the same time and with the same speed

Free Fall

downwardy verticall,)( gfreefalla

g = 9.8m/s2

If we choose the y-axis to point vertically up

gfreefalla )(

Example 2.16 A falling rockA rock is released from rest at the top of 100-m-tall building.How long does the rock take to fall to the ground, what is impact velocity?

• Y0=100m Y1=0m

• Vy0= 0 m/s t0 = 0 s

201 2/1 gtyy

sgg

yyt 52.4

)0100(2)(2 10

smgtVV yy /3.4452.48.901

Motion on an inclined plane

sin|| gas

Instantaneous Acceleration

dt

dV

t

Va

t

0

lim

tf

ti

if adtVV

Homework 2.50

• A 1000Kg weather rocket is launched straight up. The rocket motor provides a constant acceleration for 16s, then the motor stops. The rocket altitude 20 s after launch is 5100m. You can ignore the air resistance

a) What was the rocket’s acceleration during the first 16 s.

The rocket launched with Vo = 0, after 16 s

b) After motor stops, the acceleration is –g as free fall.

c) The rocket’s speed as it passes through a cloud 5100m above the ground

aatV 1601 aaty 1282/10 21

2212112 48.92/1416128))(2/1( aattgtvyy

2/2751004.78192 smaa

smattgVV /39248.916)( 1212

Quiz questions:Two stones are release from rest at certain height one after the other

• A) Will the difference in their speed increase, decrease or stay the same

B) Will their separation distance increase, decrease or stay the same