Momentum

and Collisions

Objectives: You Should Be Able To:

• Define and give examples of impulse and

momentum along with appropriate units.

• Write and apply a relationship between

impulse and momentum in one dimension.

• State the law of conservation of momentum

and apply it to the solution of problems.

Slide 9-3

What is Momentum?

• Momentum (p) is product of

mass and velocity

• p = mv

• Units: kg m/s

• Momentum is a vector quantity

that points in the same direction

as the velocity vector:

Slide 9-4

Momentum and the Impulse-Momentum Theorem

© 2015 Pearson Education, Inc.

What causes a change in

momentum (p)?

• Dropping a ball

– During drop?

– Collision with surface?

IMPULSE (J)

J = Fave t Impulse (J) is a force F

acting for an interval t.

The impulse is a vector

quantity, pointing in the

direction of the average

force vector

Units: N•s

Normally, a force acting for a short interval is not constant.

Slide 9-7

Finding the impulse on a bouncing ball

A rubber ball experiences the

force shown in FIGURE 9.4

as it bounces off the floor.

a. What is the impulse on the

ball?

b. What is the average force

on the ball?

© 2015 Pearson Education, Inc.

m m

iv

F

tavv if

m

F

fv

if mvmvtF

ImpulsetΔF tm

Fvv if

Impulse

Proof of Link between Impulse and Momentum

t

A block moving with an initial velocity vi is acted

on by a constant force F for a time t.

Impulse – Momentum theorem

When an impulse is applied to an object it changes it’s momentum

Slide 9-10

QuickCheck 9.6 Two 1.0 kg stationary cue balls are struck by cue sticks. The

cues exert the forces shown. Which ball has the greater final

speed?

A. Ball 1

B. Ball 2

C. Both balls have the same final speed.

© 2015 Pearson Education, Inc.

Slide 9-11

A light plastic cart and a heavy steel cart are both pushed

with the same force for 1.0 s, starting from rest. After the

force is removed, the

momentum of the light

plastic cart is ________ that

of the heavy steel cart.

A. Greater than

B. Equal to

C. Less than

D. Can’t say. It depends on how big the force is.

© 2015 Pearson Education, Inc.

Same force, same time same impulse

Same impulse same change of momentum

Momentum and Impulse

t

pF ave

)(

Momentum and Impulse

The rate in change of momentum is

equal to the net force applied to it

i.e. the more momentum an object has the harder

it is to stop

Or the longer to change to momentum the lower

the force!

ptF ave

https://www.youtube.com/watch?v=YY6IMLOuaSo

When you jump off of some high thing, why is it better to bend

your legs?

In which case is the impulse greater?

A) Floor

B) Foam

C) the same

In which case is the average force greater

A) Floor

B) Foam

C) the same

You drop an egg onto 1) the floor

2) a thick piece of foam rubber. In

both cases, the egg does not bounce.

I = P

Same change in momentum

p = F t

F = p/t

Smaller t = large F

A 50-g golf ball leaves the face of the club at

20 m/s. If the club is in contact for 0.002 s,

what average force acted on the ball?

t

F mv

Bouncy vs. Sticky

• Which ball has greater change in momentum?

• In which case is the impulse higher?

A 500-g baseball moves to the left at 20 m/s striking a bat. The

ball leaves in the opposite direction at 40 m/s. What was impulse

on ball?

40 m/s

t

F 20 m/s

m = 0.5 kg

+

- +

What is the impulse if the ball stuck to the bat?

Slide 9-19

QuickCheck 9.7 You awake in the night to find that your living room is on fire.

Your one chance to save yourself is to throw something that will

hit the back of your bedroom door and close it, giving you a few

seconds to escape out the window. You happen to have both a

sticky ball of clay and a super-bouncy Superball next to your

bed, both the same size and same mass. You’ve only time to

throw one. Which will it be? Your life depends on making the

right choice!

A. Throw the Superball.

B. Throw the ball of clay.

C. It doesn’t matter. Throw either.

© 2015 Pearson Education, Inc.

Larger p more impulse to door

Impulse, Momentum & Collisions • Scenarios:

– One car moving, one at rest

– Both cars moving in same direction with outside car

moving faster (rear collision)

– Both cars moving toward each other (head on collision)

• Do all scenarios for:

– Cars of equal mass AND unequal mass

• Velcro sides together

• Magnet sides together

• In Lab Notebook Record: – Scenario (total of 12)

– Observations in terms of change in velocities of each car

Impulse & Momentum in a Collision

A B vA vB

A B v’A v’B

B -FBonAt FAonB t

FAonBt = -FBonAt

pB= -pA

Rearranging: piA + piB = pfA + pfB

During a collision:

• The forces between the masses are

equal but opposite (Newton’s 3rd Law)

• Time of interaction also same

• Hence: Impulse same for each

Conservation of Momentum

• The total momentum of an isolated system

does not change.

– An isolated system is a system with no net

external force acting on it

– External forces are forces from agents outside

the system and can change the momentum of the

system.

Slide 9-23

It Depends on the System

• The goal is to choose

a system where

momentum will be

conserved.

• For a skateboarder, if

we choose just the

person, there is a net force on the system.

• If we choose the system to be the person and the cart, the

net force is zero and the momentum is conserved.

© 2015 Pearson Education, Inc.

Slide 9-24

Explosions • An explosion is when the

particles of the system move

apart after a brief, intense

interaction.

• An explosion is the opposite of a

collision.

• The forces are internal

forces total momentum is

conserved.

• Initial & final momentum

must equal zero

An 87-kg skater B collides with a 22-kg skater A initially at rest

on ice. They move together after the collision at 2.4 m/s. Find the

velocity of the skater B before the collision.

A B

vB = ? vA = 0

22 kg 87 kg

Slide 9-27

Inelastic Collisions

• A perfectly inelastic

collision is a collision in

which the two objects stick

together and move with a

common final velocity.

Slide 9-28

Recoil speed of a rifle

A 30 g ball is fired from a

1.2 kg spring-loaded toy rifle

with a speed of 15 m/s. What

is the recoil speed of the rifle?

NOTE: As the ball moves down the barrel,

there are complicated forces exerted on the

ball and on the rifle. However, if we take the

system to be the ball rifle, these are internal

forces that do not change the total

momentum.

© 2015 Pearson Education, Inc.

Slide 9-30

QuickCheck 9.10 The two boxes are on a frictionless surface. They had been

sitting together at rest, but an explosion between them has

just pushed them apart. How fast is the 2-kg box going?

A. 1 m/s

B. 2 m/s

C. 4 m/s

D. 8 m/s

E. There’s not enough information to tell.

© 2015 Pearson Education, Inc.

Slide 9-31

Collision and Momentum in 2 Dimensions

• When collisions occur in

two dimensions, we must

solve for each component

pix = pfx

piy = pfy

• In this example, the initial momentum is from m1 and

all in x-direction.

• After collision

• the momentum of both balls in x-direction must add to

equal the initial momentum

• The momentum of the balls in the y-direction must

cancel (because no initial momentum in y-direction)

Slide 9-32

2 pool balls, each of mass 200 g collide as shown below. Calculate the momentum of the white ball after the collision

Slide 9-33

Two pucks of equal mass 100 g collide on an air hockey table. Neglect

friction.

Prior to the collision, puck 1 travels in a direction that can be

considered the +x-axis at 1 m/s, and puck 2 travels in the –y-direction

at 2 m/s prior to the collision.

After the collision, puck 2 travels 30 degrees above the +x-direction

(between +x and +y) at 0.8 m/s.

What is the velocity (direction and speed) of puck 1 after the

collision?