Momentumand

Impulse

1

Momentum

What does it mean to have momentum?

2

Sports Example• Coach: “We have all the momentum.

Now we need to use that momentum and bury them in the third quarter.”

• What does the coach mean?• His team is on the move and will take

some effort to stop.

3

Momentum• All objects in motion have

momentum!• What factors determine how much

momentum an object has?

4

Linear Momentum• Linear momentum (p) – product of an object’s

mass and velocity.

• Momentum is a vector that points in the same direction as the velocity.

• Units:

vmp

kg *m/s

5

• Example #1: A freight train moves due north with a speed of 1.4 m/s. The mass of the train is 4.5 x 105 kg. How fast would an 1800 kg automobile have to be moving due north to have the same momentum?

ptrain = mtrainvtrain

=(4.5 x 105 kg)(1.4 m/s)

= 630,000 kg*m/s

pauto = mautovauto

auto

autoauto m

pv

kgs

mkg

1800

000,630

= 350 m/s

6

• Forces are not always constant. Many forces change as they are applied over a period of time.

When a baseball hits a bat, the ball is in contact with the bat for a short time interval. The force reaches its maximum value towards the middle of the time interval.

Varying Forces

t0 tf

Average Force (F)

7

Impulse• For a baseball to be hit well, both the size of the

force and the time of contact (Follow through!) are important.

• Impulse (J) – product of the average force F and the time interval Δt over which the force acts.

• Units: Newton * seconds (Ns)• Impulse is a vector that points in the same

direction as the average force.

tFJ

8

Impulse-Momentum Theorem

• Remember Newton’s 2nd Law:

• And acceleration is defined as

• Substitute:

• Rearrange:

amF

t

va

t

vmF

vm ΔΔtF

Impulse-Momentum Theorem

9

Impulse-Momentum Theorem

• Another way to write the theorem is:

pJ

10

• Example #2: A volleyball is spiked so that its incoming velocity of +4.0 m/s is changed to an outgoing velocity of -21 m/s. The mass of the volleyball is 0.35 kg. What impulse does the player apply to the ball?

v0 = +4.0 m/s

vf = -21 m/s

m = 0.35 kg pJ

0mvmv f

)/0.4)(35.0()/21)(35.0( smkgsmkg Ns 75.8

11

Impulse

Why doesn’t the egg break??Why doesn’t the egg break??

http://www.youtube.com/watch?v=7RSUjxiZnME

12

ImpulseAn object with 100 units of momentum must experience

100 units of impulse to be brought to a stop . This can come from any combination of Force x time.

An object with 100 units of momentum must experience 100 units of impulse to be brought to a stop .

This can come from any combination of Force x time.

As time Force13

Conservation of Momentum

• closed system – system where nothing is lost and no net external forces act.

• Law of Conservation of Momentum – the total momentum of any closed system does not change; the momentum before an interaction equals the momentum after the interaction.

fpp 0

14

Collisions in 1-DimensionInelastic

• Inelastic collision- a collision in which the objects stick together and move with one common velocity after colliding.

• 2 objects 1 object

15

Inelastic Collisions

BEFORE Collision AFTER Collision

fpp 0

Example 3: Two cars collide and become entangled. If car #1 has a mass of 2000 kg and a velocity of 20 m/s to the right, and car #2 has a mass of 1500 kg and a velocity of 25 m/s to the left, find the velocity of the system after the collision.

totalf m

vmvmv 2211

)3500(

)/25)(1500()/20)(2000(

kg

smkgsmkg

right the tom/s 71.0

ftotalvmvmvm 2211

16

Explosions• Explosion- process of one object splitting into 2

(or more) objects• Recoil- kickback; momentum opposite a projectile

• 1 object 2 objects (or more)

17

Explosions

BEFORE Explosion AFTER Explosion

fpp 0

Example 4: Starting from rest, two skaters push off each other on smooth level ice. Skater 1 has a mass of 88 kg and Skater 2 has a mass of 54 kg. Upon breaking

apart, Skater 2 moves away with a velocity of 2.5m/s to the right. Find the recoil velocity of Skater 1.

ff vmvm 22110

1

221 m

vmv f

f

) 88(

)m/s 5.2)( 54(

kg

kg

left the tom/s 1.53 OR

m/s 53.1

fftotal vmvmvm 22110

18

Collisions in 1-DimensionElastic

• Elastic collision – a collision in which the colliding objects bounce off each other.

• 2 objects stay 2 objects

19

Elastic CollisionsBEFORE Collision AFTER Collision

fpp 0

ffvmvmvmvm 22112211 00

This is the hardest type of problem since there are two pieces on both sides!

? ?

20

Example #5: A ball of mass 0.250 kg and a velocity of 5.00 m/s to the right collides head-on with a ball of mass 0.800 kg that is initially at rest. No external forces act on the balls. If the collision is elastic, and the 0.800 kg ball leaves the collision with a speed of 2.38 m/s, what is the velocity of the other ball after the collision?

21

Elastic CollisionsBEFORE Collision AFTER Collision

fpp 0

ffvmvmvmvm 22112211 00

If there are 2 unknowns we need to use Conservation of Energy!

? ?

22

Is Kinetic Energy Conserved?

•Elastic Collision: KE is conserved

•Inelastic Collision: KE is NOT conserved

23

Elastic Collisions with Energy• Example: A ball of mass 0.250 kg and a velocity of 5.00 m/s to the right

collides head-on with a ball of mass 0.800 kg that is initially at rest. No external forces act on the balls. If the collision is elastic, what are the velocities of the balls after the collision?

v1f = -2.62 m/sv2f = +2.38 m/s

ffvmvmvmvm 22112211 00

)/38.2)(80.0()25.0()0)(80.0()/5)(25.0( 1 smkgvkgkgsmkg

f

fpp 0

24

Elastic Collisions with Energy

• A cue ball of mass 0.165 kg is at rest on a frictionless pool table. The ball is hit dead center by a pool stick which applies an impulse to the ball causing it to roll at a velocity of 9.09 m/s. The ball then makes an elastic head-on collision with a second ball of mass 0.140 kg that is initially at rest. Find the velocities of each ball after colliding.

25

Remember Vectors???• Momentum is a vector, it follows all rules of

vectors that we have previously learned.

A

BA

B

How do we solve for the Resultant vector?

Pythagorean TheoremTrig Function (tan) to find angle

A

CB

CBA

Break into ComponentsAdd x components, Add y componentsPythagorean TheoremTrig Function

26

2-D Collisions

27

2-D Collisions

28

2-D Collisions: Inelastic• Example 1: A 200 kg car going 2 m/s east collides with a 100 kg car moving

3 m/s north. If they lock bumpers, how fast and in what direction will they be going after the collision?

29

2-D Explosion• A fireworks rocket is moving at a speed of 45.0 m/s. The

rocket suddenly breaks into two pieces of equal mass, which fly off with velocities v1 and v2, as shown in the drawing. What is the magnitude of (a) v1 and (b) v2?

30

2-D Collisions: Elasticv01 = 0.900 m/sm1 = 0.150 kg

50.0°

35.0°v02 = 0.540 m/sm2 = 0.260 kg

vf2 = 0.700 m/s

vf1

Θ

31

v01 = 0.850 m/sm1 = 0.230 kg

35.0° 15.0°

v02 = 0.430 m/sm2 = 0.190 kg

vf1 = 0.600 m/s

vf2

Θ

Find the missing values: vf2 and θ

32