1.MOMENTUM l Momentum = mass times velocity Units - kg m/s.

1. MOMENTUM

Momentum = mass times velocity

vmp

Units - kg m/s

2. IMPULSE

Collisions involve forces (there is a v).

Impulse = force times time.

tFI

Units - N s

3. IMPULSE CHANGES MOMENTUM

Impulse = change in momentum

amF

vmtF

t

vmF

vmtF

pI

)( if vmvmtF

)( if vvmtF

)( if pptF

Case 1Increasing Momentum

Follow through

Examples:Long Cannons

Driving a golf ballCan you think of others?

t t p

I

F

I

p

Case 2Decreasing Momentum over a Long Time

Examples:

Bungee Jumping

Can you think of others?

Ip

tF

tF

Warning – May be dangerous

Case 3Decreasing Momentum over a Short Time

Examples:

Boxing (leaning into punch)

Head-on collisions

Can you think of others?

tFIp

5. CONSERVATION OF MOMENTUM

Example:Rifle and bullet

Demo - Rocket balloon Demo - Clackers

Video - Cannon recoilVideo - Rocket scooter

Consider two objects, 1 and 2, and assume that no external forces are acting on the system composed of these two particles.

ifvmvmtF

11111

Impulse applied to object 1

ifvmvmtF

22222

ififvmvmvmvm

222211110

Impulse applied to object 2

Total impulseappliedto system

ffiivmvmvmvm

22112211

or

Apply Newton’s Third Law21

FF

tFtFor 21

Internal forces cannot cause a change in momentum of the system.

For conservation of momentum, the external forces must be zero.

6. COLLISIONS

Collisions involve forces internal to

colliding bodies.

Elastic collisions - conserve energy and

momentum

Inelastic collisions - conserve momentum

Totally inelastic collisions - conserve

momentum and objects stick together

Elastic and Inelastic Collisions

In an elastic collision the total kinetic energy is conserved Momentum is conserved in any

collision Example:

What are signs of final velocities?

22

212

1223

21

2112

2

and 22Let

ff

ff

ii

mvmvmvK

mmm

mmm

vvvP

vvv

Elastic and Inelastic Collisions

Example (cont.): Consider reference frame where CM is at rest

vvvvvvvv

vvvv

vv

vvvvP

vvvvvvvv

vvvv

31Lab

CMCM2235Lab

CMCM11

32

CM234

CM1

CM*CM*

CM2CM1CM2CM1

32Lab

CM2CM234Lab

CM1CM1

31

31Lab

CM

,

,

220

and ,

)2(

ffff

ff

if

iiii

iiii

m

mm

--

mm

Elastic and Inelastic Collisions

In an inelastic collision the total kinetic energy is not conserved Momentum is conserved

in any collision Example: case where

particles stick together

vv

vvvvP

vvv

31

21

2112

32

and 22Let

f

fff

ii

mmmm

mmm

Elastic and Inelastic Collisions

Example: Ballistic Pendulum

ghv

ghmmvmm

vv

vmmvm

mmm

A

B

Ammm

B

BA

2

)()(

)(

1

21

21

1

1

212

2121

1

2111

Collision between two objects of the same mass. One mass is at rest.

Collision between two objects. One not at rest initially has twice the mass.

Collision between two objects. One at rest initially has twice the mass.

Simple Examples of Head-On Collisions

(Energy and Momentum are Both Conserved)

Head-On Totally Inelastic Collision Example

Let the mass of the truck be 20 times the mass of the car.

Using conservation of momentum, we get

mphvtruck 60 mphvcar 60

vmmphmmphm )21()60()60(20

vmph 21)60(19

)60(21

19mphv

mphv 3.54

initial momentum of system = final momentum of system

Remember that the car and the truck exert equal but oppositely directed forces upon each other.

What about the drivers? The truck driver undergoes the same

acceleration as the truck, that is

t

mph

t

mph

7.5)603.54(

The car driver undergoes the same acceleration as the car, that is

t

mph

t

mphmph

3.114)60(3.54

The ratio of the magnitudes of these two accelerations is

207.5

3.114

Remember to use Newton’s Second Law to see the forces involved.

For the truck driver his mass times his acceleration gives

F

am

For the car driver his mass times his greater acceleration gives

ma

F

, big trucks that is. Your danger is of the order of twenty times

greater than that of the truck driver.

TRUCKS Don’t mess with T

Collisions in 2D

Use Conservation of Momentum in each direction Consider case where

one particle is at rest

In CM frame particles are back-to-back!

sinsin

coscos

2211

221111

ff

ffi

vmvm

vmvmvm

Rocket Propulsion

“Rockets can’t fly in vacuum. What do they have to push against?” Nonsense. Rockets

don’t push; they conserve momentum, and send parts (fuel) away from the body as fast as possible

Rocket Propulsion

How fast do rockets accelerate? Start at rest, with mass

M+m Some time t later, have

expelled m at speed ve, to conserve momentum, rest of rocket (M) must have velocity (in the other direction) of

v = ve m/M

Rocket Propulsion

How fast do rockets accelerate?

Thrust: (instantaneous) force on rocket

i

ff

i

f

i

e

e

m

m

e

m

m mdm

e

t

t dtdm

mv

if

dtdm

mv

dtdv

vvdtvv

a

ln

dtdm

evF