Momentum

Linear Momentum

• A vector quantity defined as the product of an object’s mass and velocity

• p = momentum

p = mass X velocity

Linear Momentum

•What does the term momentum mean to you?–What are examples of momentum?

Example 1

What velocity must a car with a mass of 1210 kg have in order to have a momentum of 5.6 X 104 kg·m/s to the east?

46 m/s to the east

Example 2

An ostrich with a mass of 146 kg is running to the right with a velocity of 17 m/s. Find the momentum of the ostrich.

2.5 X 10 3 kg·m/s to the right

Example 3

A 21 kg child is riding a 5.9 kg bike with a velocity of 4.5 m/s to the north west.

What is the total momentum of the child and the bike together?

1.2X 10 2 kg·m/s to the northwest

Example 4

A 21 kg child is riding a 5.9 kg bike with a velocity of 4.5 m/s to the north west.

What is the momentum of the child?

94 kg·m/s to the northwest

Example 5

A 21 kg child is riding a 5.9 kg bike with a velocity of 4.5 m/s to the north west.

What is the momentum of the bike?

27 kg·m/s to the northwest

Impulse-Momentum Theorem

• F = ma

But F = ∆p/∆t,

Force = change in momentum

time interval

Impulse-Momentum Theorem

• F∆t = ∆p

or

• F∆t = ∆p = mvf- mvi

Force x time interval = change in momentum

Impulse-Momentum Theorem

• Impulse- constant external force, product of the force and the time over which it acts on an object

• “Follow through”

Example 6

A .50 kg football is thrown with a velocity of 15 m/s to the right. A stationary receiver catches the ball and brings it to rest in 0.020 seconds. What is the force exerted on the receiver?

380 N to the right

Example 7

A 0.40 kg soccer ball approaches a player horizontally with a velocity of 18 m/s to the north. The player strikes the ball and causes it to move in the opposite direction with a velocity of 22 m/s. What impulse was delivered to the ball by the player?

16 kg· m/s to the south

Example 8

A 0.50 kg object is at rest. A 3.00 N force to the right acts on the object during a time interval of 1.5 s.

What is the velocity of the object at the end of this interval?

9.0 m/s to the right

Example 9

A 0.50 kg object is at rest. A 3.00 N force to the right acts on the object during a time interval of 1.5 s.

At the end of this interval, a constant force of 4.00 N to the left is applied for 3.00 s. What is the velocity at the end of the 3.00 s?

15 m/s to the left

Impulse-Momentum Theorem

• Stopping times and distances are influenced by the Impulse-Momentum Theorem

• When thinking of a car and truck traveling at the same speed, you know that if the force of the brakes are applied equally for both vehicles. The car will stop in a shorter distance.

Example 11

A 2500 kg car traveling to the north is slowed down uniformly from an initial velocity of 20.0 m/s by a 6250 N braking force acting opposite the car’s motion. Use the impulse-momentum theorem to answer the following questions:

What is the car’s velocity after 2.50 s?

14 m/s to the north

Example 12

A 2500 kg car traveling to the north is slowed down uniformly from an initial velocity of 20.0 m/s by a 6250 N braking force acting opposite the car’s motion. Use the impulse-momentum theorem to answer the following questions:

How far does the car move during 2.50 s?

42 m

Example 13

A 2500 kg car traveling to the north is slowed down uniformly from an initial velocity of 20.0 m/s by a 6250 N braking force acting opposite the car’s motion. Use the impulse-momentum theorem to answer the following questions:

How long does it take the car to come to a complete stop?

8.0 s

Impulse-Momentum Theorem

• Change in momentum over longer time requires less force

Conservation of Momentum

• The total momentum of all objects interacting with one another remains constant regardless of the nature of the forces between the objects.

Conservation of Momentum

Situation:

Ball A is hit and collides with a stationary ball B.

• Before the collision ball B has a momentum equal to zero.

• After the collision ball B gains momentum and ball A loses momentum.

Conservation of Momentum

Conservation of Momentum

The total momentum of the two balls together remains constant!!!!!!

• Therefore:– Pinitial = Pfinal

–M1V1i + M2V2i = M1V1f + M2V2f

Conservation of Momentum

Momentum is conserved in collisions

Conservation of Momentum

Momentum is conserved for objects pushing away from each other.

An example is jumping into the air.

upward = downward momentum momentum

Example 14

A 63 kg astronaut is on a spacewalk when the tether line to the shuttle breaks. The astronaut is able to throw a 10 kg oxygen tank in a direction away from the shuttle with a speed of 12.0 m/s, propelling the astronaut back to the shuttle. Find the final speed of the astronaut who was initially at rest

1.90 m/s

Example 15

A 63 kg astronaut is on a spacewalk when the tether line to the shuttle breaks. The astronaut is able to throw a 10 kg oxygen tank in a direction away from the shuttle with a speed of 12.0 m/s, propelling the astronaut back to the shuttle. What is the maximum distance the astronaut can be from the craft when the line breaks in order to return to the craft within 60 seconds

114 meters

Example 16

An 85 kg fisherman jumps from a dock into a 135 kg rowboat at rest on the west side of the dock. If the velocity of the fisherman is 4.30 m/s to the west as he leaves the dock, what is the final velocity of the fisherman and the boat?

1.66 m/s to the west

Example 17

Each ball has a mass of .50 kg. The green ball, traveling at 12.0 m/s, strikes the blue ball, which is at rest. Assuming that all collision are head-on, find the final speed of the blue ball.

A. The green ball stops moving after it strikes the blue ball.

12.0 m/s

Example 18

Each ball has a mass of .50 kg. The green ball, traveling at 12.0 m/s, strikes the blue ball, which is at rest. Assuming that all collision are head-on, find the final speed of the blue ball.

B. The green ball continues moving after the collision at 2.4 m/s in the same direction.

9.6 m/s

Collisions

Perfectly inelastic collisions–When two objects stick together and

move as one mass

Elastic collisions

–When two objects return to their original shape and move separately

Perfectly Inelastic Collisions

The initial momentum of the 2 objects equals the final momentum of the objects combined

M1V1i + M2V2i = (M1 + M2) Vf

Example 19

1500 Kg car traveling at 15.0 m/s to the south collides with a 4500 kg truck that is initially at rest at a stoplight. The car and truck stick together and move together after the collision. What is the final velocity of the two vehicle mass?

3.8 m/s to the south

Example 20

A grocery shopper tosses a 9.0 kg bag of rice into a stationary 18.0 kg cart. The bag hits the cart with a horizontal speed of 5.5 m/s towards the front of the cart. What is the final speed of the cart and the bag?

1.8 m/s

Example 21

A 15,000 kg railroad car moving at 7.00 m/s to the north collides with and sticks to another railroad car of the same mass that is moving in the same direction at 1.50 m/s. What is the velocity of the joined cars after the collision?

4.0 m/s to the north

Example 22

A dry cleaner throws a 22 kg bag of laundry onto a stationary 9.0 kg cart. The cart and laundry bag begin moving at 3.0 m/s to the right. Find the velocity of the laundry bag before the collision.

4.2 m/s to the right

Perfectly Inelastic Collisions

The total kinetic energy doesn’t remain constant when objects collide and stick together.

Energy is converted into other forms.

Objects are deformed and lose some kinetic energy.

Example 23

A 0.25 kg arrow with a velocity of 12 m/s to the west strikes and pierces the center of a 6.8 kg target.

A. What is the final velocity of the combined mass?

.43 m/s to the west

Example 24

A 0.25 kg arrow with a velocity of 12 m/s to the west strikes and pierces the center of a 6.8 kg target.

B. What is the decrease in kinetic energy during the collision?

17 J

Example 25

During practice, a student kicks a 0.40 kg soccer ball with a velocity of 8.5 m/s to the south into a 0.15 kg bucket lying on its side. The bucket travels with the ball after the collision.

A. What is the final velocity of the combined mass?

6.2 m/s to the south

Example 26

During practice, a student kicks a 0.40 kg soccer ball with a velocity of 8.5 m/s to the south into a 0.15 kg bucket lying on its side. The bucket travels with the ball after the collision.

B. What is the decrease in kinetic energy during the collision?

3 J

Elastic Collisions

The total kinetic energy is conserved.

M1V1i + M2V2i = M1V1f + M2V2f

Example 27

A 0.015 kg marble moving to the right at 22.5 cm/s makes an elastic head-on collision with a 0.015 kg marble moving to the left at 18.0 cm/s. After the collision, the first marble moves to the left at 18.0 cm/s.

A. Find the velocity of the second marble after the collision.

22.5 cm/s to the right

Example 28

A 0.015 kg marble moving to the right at 22.5 cm/s makes an elastic head-on collision with a 0.015 kg marble moving to the left at 18.0 cm/s. After the collision, the first marble moves to the left at 18.0 cm/s.

B. Verify your answer by calculating the total kinetic energy before and after the collision.

.00062 J

Example 29

A 16.0 kg canoe moving to the left at 12 m/s makes an elastic head-on collision with a 4.0 kg raft moving to the right at 6.0 m/s. After the collision, the raft moves to the left at 22.7 m/s.

B. Find the velocity of the canoe after the collision.

5 m/s to the left

Example 30

A 16.0 kg canoe moving to the left at 12 m/s makes an elastic head-on collision with a 4.0 kg raft moving to the right at 6.0 m/s. After the collision, the raft moves to the left at 22.7 m/s.

B. Verify your answer by calculating the total kinetic energy before and after the collision.

Kei= 1300 J and Kef= 1200 J

Inelastic Collisions

Most collisions are neither elastic or perfectly inelastic.

– They lose some kinetic energy

– Deformation takes place

– Objects move together for a period of time, then move separately.

Example 31

A 82 kg man drops from rest on a diving board 3.0 m above the surface of the water and comes to rest 0.55 s after reaching the water. What force does the water exert on him?

1100 upward