Chapter 8Chapter 8

Impulse and MomentumImpulse and Momentum

THE LINEAR MOMENTUMTHE LINEAR MOMENTUM

Momentum = mass times velocity “Think of it as inertia in motion”

vmp

Units - kg m/s or sl ft/s

AN IMPULSEAN IMPULSE

Collisions involve forces (there is a v).

Impulse = force times time.

tFI

Units - N s or lb s

AN IMPULSE CAUSES A CHANGE AN IMPULSE CAUSES A CHANGE MOMENTUMMOMENTUM

Impulse = change in momentum

amF

vmtF

t

vmF

vmtF

pI

)( if vmvmtF

)( if vvmtF

)( if pptF

Case 1Case 1Increasing MomentumIncreasing Momentum

Follow through

Examples:Long Cannons

Driving a golf ballCan you think of others?

pItF

Case 2Decreasing Momentum over a Long Time

Examples:

Rolling with the Punch

Bungee Jumping

Can you think of others?

Ip

tF

tF

Warning – May be dangerous

Case 3Case 3Decreasing Momentum over a Short TimeDecreasing Momentum over a Short Time

Examples:

Boxing (leaning into punch)

Head-on collisions

Can you think of others?

tFIp

BOUNCINGBOUNCING

There is a greater impulse with bouncing.

Example:Pelton Wheel

Pelton Wheel Water Sprinkler

Consider a hard ball and a clay ball that have +10 units of momentum each just before hitting a wall.

The clay ball sticks to the wall and the hard ball bounces off with -5 units of momentum.

Which delivered the most “punch” to the wall?

Initial momentum of the clay ball is 10.Final momentum of clay ball is 0.The change is 0 - 10 = - 10.It received - 10 impulse so itapplied + 10 to the wall.

Initial momentum of the hard ball is 10.Final momentum of hard ball is - 5.The change is - 5 - 10 = - 15.It received - 15 impulse so itapplied + 15 to the wall.

Example:Rifle and bullet

Demo - Rocket balloon Demo - Clackers

Video - Cannon ShootVideo – Scooter Propulsion

CONSERVATION OF LINEAR CONSERVATION OF LINEAR MOMENTUMMOMENTUM

IN COLISIONS AND EXPLOSIONSIN COLISIONS AND EXPLOSIONS

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

11111 umvmtF

Impulse applied to object 1

22222 umvmtF

222211110 umvmumvm

Impulse applied to object 2

Total impulseappliedto system

22112211 vmvmumum

or

Apply Newton’s Third Law21

FF

tFtFor 21

In one dimension in component form,In one dimension in component form,

xxxx vmvmumum 22112211

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

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

IN COLLISIONS AND IN COLLISIONS AND EXPLOSIONSEXPLOSIONS

Collisions involve forces internal to colliding bodies.

Inelastic collisions - conserve momentum Totally inelastic collisions - conserve

momentum and objects stick together

A PERFECTLY ELASTIC A PERFECTLY ELASTIC COLLISIONCOLLISION

2222

12112

12222

12112

1 vmvmumum

Perfectly elastic collisions - conserve energy and momentum

Demos Demos

Demo - Momentum ballsDemo - Momentum balls

Demo - Small ball/large ball dropDemo - Small ball/large ball drop

Demo - Funny Balls Demo - Funny Balls

Head-On Totally Inelastic Head-On Totally Inelastic Collision ExampleCollision 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

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

For the truck driver his mass times his acceleration gives

F

am

ma

F For the car driver his mass times his greater acceleration gives

Your danger is of the order of twenty times

greater than that of the truck driver.

TRUCKS. Don’t mess with T

COEFFICIENT OF COEFFICIENT OF RESTITUTIONRESTITUTION

For any collision between two bodies moving along a single straight line, the coefficient of restitution e is defined as

xx

xx

uu

vve

21

12

u’s are velocities before impact.v’s are velocities after impact.For perfectly elastic collisions e = 1. For inelastic collisions e < 1.For totally inelastic collisions e = 0.

xx

xx

uu

vve

21

12

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)

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

Example of Non-Head-On Collisions

(Energy and Momentum are Both Conserved)

If you vector add the total momentum after collision,you get the total momentum before collision.

THE CENTER OF MASSTHE CENTER OF MASS

The center of mass of an object of mass m is the single point that moves in the same way as a point mass would move when subjected to the same external forces that act on the object.

m

Facm

The coordinates of the center of mass areThe coordinates of the center of mass are

i

iicm m

mzz

i

iicm m

mxx

i

iicm m

myy

.

CENTER OF MASS ANDCENTER OF MASS ANDCENTER OF GRAVITYCENTER OF GRAVITY

Center of mass - average position of mass

Earth

.

Center of gravity - average position of weight

Path of center of mass of a rotating object will

be a straight line if no external forces act on

the object.

Locating the Center of GravityLocating the Center of Gravity

Demo - Meterstick

Demo - Map of Texas

Demo - Balancing eagle

Demo - Curious George

Center can be outside of the object.

Examples: high jump and pole vaulting