Momentum

Inertia in Motion

Definition of Momentum• Momentum is the product of mass and

velocity.

p mv

Momentum mskg Mass kg Velocity m

s

Momentum Depends on Velocity• An object must be moving in order to have

momentum.• Consider the truck at rest.• Consider the skate boarder. He has momentum.

0Truckp

0Skateboarderp

Momentum Depends on Mass• Larger objects traveling at the same velocity as

smaller objects will have greater momentum.

ManTruck pp

The Nature of Momentum• Momentum is a vector quantity.

– It has magnitude (how big).– It has direction (right(+) or left(-)).

• Direction is denoted by sign (+ or -).

0Truckp 0Busp

Collisions• The collisions that we will study will involve two objects.• Each object possesses its own momentum.• Collisions are broken down into two categories:

– Elastic (Objects Separate)– Inelastic (Objects Stick Together)

Momentum – Elastic Collisions• Elastic Collision – a collision in which the colliding bodies

do not stick together.• The equation used for elastic collisions is as follows.

FFII vmvmvmvm 22112211

Momentum – Inelastic Collisions• Inelastic Collision – a collision in which the colliding bodies

stick together.• The equation used for inelastic collisions is as follows.

1 1 2 2 Fm v m v MV

Elastic Collisions• The total momentum of the system remains

constant.

• Elastic collision, objects move off with separate velocities.

• This means that they can bounce.• Or they can be projected from rest (like a cannon).

Head on Elastic collisions

( ')( ') ( ' )( ' )a a a a a b b b b bm v v v v m v v v v

' 'a a b b a a b bm v m v m v m v

' 'a a a a b b b bm v m v m v m v

( ') ( ' )a a a b b bm v v m v v

In Elastic collisions

Head on Elastic collisions

( ')( ') ( ' )( ' )

( ') ( ' )a a a a a b b b b b

a a a b b b

m v v v v m v v v v

m v v m v v

( ') ( ' )a a b bv v v v

In Elastic collisions, KE is conserved

2 2 2 21 1 1 12 2 2 2' 'a a a a b b b bm v m v m v m v

2 2 2 21 1 1 12 2 2 2' 'a a b b a a b bm v m v m v m v

2 2 2 2( ' ) ( ' )a a a b b bm v v m v v 2 2( ) ( )( )a b a b a b

Impulse (mv)• When objects collide, their velocities

change. This means that their individual momentums will also change.

• Impulse simply defines the change in momentum for an individual object (denoted by mv).

Impulse (mv)• There are two ways to calculate impulse:

– You can subtract initial momentum from final momentum.

– Or you can take the product of force and time.

• The units for impulse are in Ns, which is equal to kgm/s

After BeforeJ mv p p

J mv Ft

End Chapter 7

Sample Problem 1 (Impulse)• A golf ball, initially at rest, is struck by a

golf club.

– The golf ball (m = 0.1kg) rolls away with a velocity of 35m/s.What is the initial momentum of the ball?What is the final momentum of the ball?What is the impulse on the ball?

0.1 0 0m ms sBefore Ip mv kg kg 0.1 35 3.5m ms sAfter Fp mv kg kg

After Beforep p p

3.5 0 3.5m m ms s sp kg kg kg

03.53.5

Swing

Practice Problem (Impulse)• A soccer ball (m = 0.4kg) approaches a player with a

velocity of 13m/s (left). The player kicks the ball giving it a new velocity of 20m/s (right).What is the initial momentum of the soccer ball?What is the final momentum of the soccer ball?What is the impulse felt by the soccer ball?

0.4 13 5.2m ms sp mv kg kg 0.4 20 8.0m m

s sp mv kg kg

8.0 5.2

13.2

F I

m ms s

ms

p p p

p kg kg

p kg

Kick

5.2 msIp kg

8.0 msFp kg

Ballistic pendulum• The ballistic pendulum has two parts.

Conservation of MomentumConservation of Energy

Ballistic pendulum

0 ( ) 'mv m M v

212 ( ) ' ( )m M v m M gh

( ) 'm M vv

m

Conservation of Momentum

Conservation of Energy

KE GPE21

2 'mv mgh

' 2v gh

( )2

m Mv gh

m

Sample Problem (Elastic)

FFII vmvmvmvm 22112211

mcannon = 300kg

mball = 10kg

Fire!

Momentum

• Identify the number and types of collisions in the animation below.

Practice Problem 3 (Elastic)• Two air track cars collide and bounce.• The red car is initially traveling at a velocity of 0.50m/s to the right

while the yellow car is traveling at a velocity of 1.5m/s to the left.• After the collision the red car is moving 2.0m/s to the left with the

yellow car following, but not touching.

Air Track0.3kg 0.8kg

mred = 0.3kg

myellow = 0.8kgFFII vmvmvmvm 22112211

What is the initial momentum of the red car?What is the initial momentum of the yellow car?What is the final momentum of the red car?What is the final momentum of the yellow car?

What is the final velocity of the yellow car?

0.3 0.5 0.15m ms sp mv kg kg 0.3 2.0 0.6m m

s sp mv kg kg 0.8 1.5 1.2m ms sp mv kg kg

0.15 0.61.2

0.15 1.2 0.6 ?

? 0.45 mskg

2

0.45

0.8

ms

F

kgv

kg

2 0.5625m

sFv

0.45

pp mv v

m

Collide

Practice Problem 4 (Elastic)• A gun (m = 1.5kg), initially at rest, is fired sending its

bullet (m = 0.1kg) at a velocity of . 180 ms

Before

After

What is the initial momentum of the gun and bullet?What is the total momentum of the system before firing?What is the total momentum of the system after firing?What is the final momentum of the bullet?What is the final momentum of the gun?What is the final velocity of the gun?

FFII vmvmvmvm 22112211 0 0 18 18

0.1 180 18m ms sp mv kg kg

1812

1.5

ms m

s

p kgp mv v

m kg

1.5 0 0

0.1 0 0

msGun

msBullet

p mv kg

p mv kg

00

0 0 18

18

x

x

Homework

• WS1 – #’s 1,2, 4-17

Inelastic Collisions• The Law of Conservation of Momentum: that the total

momentum of the system remains constant, no matter what the collision.

• In an inelastic collision, objects stick together.

• This means that they have the same final velocity (vF).

S

N

v2vF vF

Collide

Inelastic Collision Equation• The inelastic collision equation takes

into account the momentum of each object before and after the collision.

• Because the objects are combined after the collision, and they have the same final velocity (vF).

Before After

Inelastic Collision Equation

TotAfterTotBefore pp

FII vmmvmvm 212211

Initial Momentum Object 1

Initial Momentum Object 2

Final Combined Momentum

Objects 1 & 2

Sample Problem (Inelastic)The big rig backs up with an initial velocity of 2.0m/s and collides with the trailer, which is initially at rest.

FII vmmvmvm 212211 What is the initial momentum of the truck?What is the initial momentum of the trailer?What is the combined mass of the truck and trailer?What is the final velocity of the truck and trailer combo?

mTruck = 2000.0kg

mTrailer = 3000.0kg

003000 kgmvp sm

sm kgkgmvp 40000.22000 kgkgkgmm 50003000200021 Fs

m vkgkg 500004000

kg

vkg

kg

kg Fsm

5000

5000

5000

4000 s

mFv 8.0

04000 5000

BeforeAfter

0.8

ICE

Practice Problem 5 (Inelastic)• Happy Penguin (m = 7.0kg) slides across the ice on a magnet

with a velocity of 3.0m/s. • Cool Penguin (m = 8.0kg) comes up from behind with a velocity

of 5.0m/s. He, also having a magnet, will collide and stick.

FII vmmvmvm 212211

S

N

C

N

S

H

What is the initial momentum of Cool Penguin?What is the final momentum of Happy Penguin?What is the combined mass of the two penguins?What is the final velocity of the penguin pair?

sm

sm kgkgmvp 0.210.30.7 s

ms

m kgkgmvp 0.400.50.8 kgkgkgmm 0.150.80.721

0.15

0.15

0.15

0.61 Fv

sm

Fv 07.4

Fv0.150.61

40.021.0 15.0 4.07

Collide

Momentum While climbing a cliff, a super model (m =

51.0 kg) slips and falls. She falls for 2s before she is rescued by

Super Doctor Physics (m = 63.0 kg, v = 27.85 m/s).

What was their velocity immediately after the collision?

Homework

• WS 2– # 1,2

• WS 3– #1

Sample Problem 2 (Impulse)• A baseball is struck by a bat with a force of

3000N for a duration of 0.05s. What is the change in momentum of the baseball?

p Ft

150p Ns 3000 0.05p N s

150 msp kg

or

NF 3000st 05.0

Hit

Practice Problem 7 (Impulse)• A tennis ball is struck with a force of 400N

for a time of 0.15s.What is the impulse felt by the tennis ball?

p Ft 400 0.15 60p N s Ns

Hit

Summary• All problems surrounding momentum are

based on the following principles:– Momentum depends on both mass and velocity.– Momentum is conserved, regardless of collision.– Collision type determines the equation used.– Impulse is simply change in momentum.

Trains are stupid cuz they can’t do

Physics!