Law of Conservation of Momentum and Collisions

Chapter 8.4-8.5

Momentum is conserved for all collisions as long as external forces don’t interfere.

LAW OF CONSERVATION OF MOMENTUM

• In the absence of outside influences, the total amount of momentum in a system is conserved.

• The momentum of the cue ball is transferred to other pool balls.

• The momentum of the pool ball (or balls) after the collision must be equal to the momentum of the cue ball before the collision

• p before = p after

Whenever objects collide in the absence of external forces, the net momentum of the objects before the collision equals the net momentum of the objects after the collision.

8.5 Law of Conservation and Collisions

Motion of the cue ball Motion of the other balls

Figure 8.10Momentum of cannon and cannonball

Read Page 131

• Read 1st paragraph• What does Newton’s 3rd law have to say about the net

force of the cannon-cannonball system?

• Why is the momentum of the cannon-cannonball system equal to zero before and after the firing?

Read Page 131

• Read 1st paragraph• What does Newton’s 3rd law have to say about the net force of the

cannon-cannonball system?• The net force of this system equals zero because the

action and reaction forces cancel each other out

• Why is the momentum of the cannon-cannonball system equal to zero before and after the firing?

• The momentum in the system must be conserved; so if the system starts with zero momentum, it must end with zero momentum.

Read Page 131

• Read 2nd paragraph• Why is momentum a vector quantity?

• Explain the difference between the momentum of the cannon and the momentum of the cannonball, and the momentum of the cannon-cannonball system.

Read Page 131

• Read 2nd paragraph• Why is momentum a vector quantity?• Momentum is a quantity that expresses both

magnitude and direction.

• Explain the difference between the momentum of the cannon and the momentum of the cannonball, and the momentum of the cannon-cannonball system.

• After the firing occurs, both the cannon and cannonball have the same momentum (big mass, small velocity vs. small mass, big velocity). But since the momentum for each is moving in the opposite direction, the momentums cancel out, causing the cannon-cannonball system’s momentum to equal zero.

Read Page 131

• Read 3rd paragraph• Why do physicists use the word conserved for

momentum?

• State the law of conservation of momentum.

Read Page 131

• Read 3rd paragraph• Why do physicists use the word conserved for

momentum?• The word conserved refers to quantities

that do not change.

• State the law of conservation of momentum.• In the absence of an external force, the

momentum of a system remains the same.

Read Page 131

• What does system mean?

• In terms of momentum conservation, why does a cannon recoil when fired?

Final Thoughts about Page 131

• What does system mean?• The word system refers to a group of interacting

elements that comprises a complex whole.

• In terms of momentum conservation, why does a cannon recoil when fired?

• The cannon must recoil in order for momentum to be conserved. (The momentum of the cannon-cannonball system was zero before the firing, and must remain zero after the firing.)

Read Page 131

• What does conservation of momentum mean?

• Conservation of momentum means that the amount of momentum in a system does not change.

• Why is the momentum cannon-cannonball system equal to zero?

• The momentum of the cannonball cancels out the recoil of the cannon (both move in opposite directions with an equal amount of momentum.

The momentum before firing is zero. After firing, the net momentum is still zero because the momentum of the cannon is equal and opposite to the momentum of the cannonball.

8.4 Conservation of Momentum

Velocity cannon to left is negative

Velocity of cannonball to right is positive

(momentums cancel each other out!)

8.5 Two Types of Collisions• Elastic Collision: When objects collide without sticking together

--Kinetic energy is conserved

--No heat generated

• Inelastic Collision: When objects collide and deform or stick together.

--Heat is generated

--Kinetic energy is not conserved

Changes in Velocity Conserve Momentum

A. Elastic collisions with equal massed objects show no change in speed to conserve momentum

• http://www.walter-fendt.de/ph14e/ncradle.htm• http://www.walter-fendt.de/ph14e/collision.htmB. Elastic collisions with inequally massed objects show

changes in speed to conserve momentum– Larger mass collides with smaller mass—smaller mass object’s

speed is greater than the larger mass object– Smaller mass object collides with larger mass object—larger

mass object’s speed is much less than the smaller mass object– http://www.walter-fendt.de/ph14e/collision.htm

C. Addition of mass in inelastic collisions causes the speed of the combined masses to decrease in order for momentum to be conserved

a. A moving ball strikes a ball at rest.

8.5 Examples of Elastic Collisions when the objects have identical masses

Note: purple vector arrow represents velocity: speed and direction

a. A moving ball strikes a ball at rest.

8.5 Examples of Elastic Collisions when the objects have identical masses

Momentum of the first ball was transferred to the second; velocity is identical

b. Two moving balls collide head-on.

8.5 Examples of Elastic Collisions when the objects have identical masses

b. Two moving balls collide head-on.

8.5 Examples of Elastic Collisions when the objects have identical masses

The momentum of each ball was transferred to the other; each kept same speed in opposite direction

c. Two balls moving in the same direction at different speeds collide.

8.5 Examples of Elastic Collisions when the objects have identical masses

8.5 Examples of Elastic Collisions when the objects have identical masses

The momentum of the first was transferred to the second and the momentum of the second was transferred to the first. Speeds to conserve momentum.

c. Two balls moving in the same direction at different speeds collide.

Example of an elastic collision with objects same speed but different masses

What happens to the speed of the smaller car after the elastic collision with the more massive truck?

Notice that the car has a positive velocity and the truck a negative velocity. What is the total momentum in this system?

Example of an elastic collision with objects same speed but different masses

What happens to the speed of the smaller car after the elastic collision with the more massive truck? (the car’s speed increases to conserve momentum)

Notice that the car has a positive velocity and the truck a negative velocity. What is the total momentum in this system? (40,000 kg x m/s)

8.5 Inelastic Collisions

Start with less mass, end up with more mass

Notice how speed changes to conserve momentum (more mass, less speed—inverse relationship!)

Calculating conservation of momentum

• Equation for elastic collisions

m1v1 + m2v2 = m1v1 + m2v2

• Equation for inelastic collision

m1v1 + m2v2 = (m1 + m2)v2

Before collision

After collision

Before collision

After collision

Conservation of Momentum in an elastic collision

Before elastic collision After elastic collision

Cart A mass = 1 kgCart B mass = 1 kgCart A speed = 5 m/sCart B speed = 0 m/s

Cart A mass = 1 kgCart B mass = 1 kgCart A speed = 0 m/sCart B speed = 5 m/s

A B

Conservation of Momentum in an elastic collision

Before elastic collision After elastic collision

Cart A mass = 1 kgCart B mass = 1 kgCart A speed = 5 m/sCart B speed = -5 m/s

Cart A mass = 1 kgCart B mass = 1 kgCart A speed = -5 m/sCart B speed = 5 m/s

A B

Conservation of Momentum in an elastic collision

Before elastic collision After elastic collision

Cart A mass = 1 kgCart B mass = 5 kgCart A speed = 5 m/sCart B speed = 0 m/s

Cart A mass = 1 kgCart B mass = 5 kgCart A speed = 0 m/sCart B speed = 1 m/s

A B

Conservation of Momentum in an elastic collision

Before elastic collision After elastic collision

Cart A mass = 6 kgCart B mass = 1 kgCart A speed = 10 m/sCart B speed = 0 m/s

Cart A mass = 6 kgCart B mass = 1 kgCart A speed = 2 m/sCart B speed = 48 m/s

A B

Conservation of Momentum in an inelastic collision

Big fish mass = 4 kg

Small fish mass = 1 kg

Small fish speed = 5 m/s

Large fish speed = 0 m/s

Before inelastic collision

Big fish mass + Small fish mass =

Small fish + Large fish speed =

After inelastic collision

5 kg

1 m/s

m1v1 = v2

m1 + m2

think!One glider is loaded so it has three times the mass of another glider. The loaded glider is initially at rest. The unloaded glider collides with the loaded glider and the two gliders stick together. Describe the motion of the gliders after the collision.

8.5 Collisions

think!One glider is loaded so it has three times the mass of another glider. The loaded glider is initially at rest. The unloaded glider collides with the loaded glider and the two gliders stick together. Describe the motion of the gliders after the collision.

Answer: The mass of the stuck-together gliders is four times that of the unloaded glider. The velocity of the stuck-together gliders is one fourth of the unloaded glider’s velocity before collision. This velocity is in the same direction as before, since the direction as well as the amount of momentum is conserved.

8.5 Collisions

1. Conservation of Momentum in an elastic collision

m1v1 = v2m2

Before elastic collision

After elastic collision

Cart A mass = 1 kgCart B mass = 5 kgCart A speed = 5 m/sCart B speed = 0 m/s

Cart A mass = 1 kgCart B mass = 5 kgCart A speed = 0 m/s

Find Cart B speed

A B

2. Conservation of Momentum in an elastic collision

m1v1 = v2m2

Before elastic collision

After elastic collision

Cart A mass = 5 kgCart B mass = 2 kgCart A speed = 10 m/sCart B speed = 0 m/s

Cart A mass = 5 kgCart B mass = 2 kgCart A speed = 0 m/s

Find Cart B speed

A B

Consider a 6-kg fish that swims toward and swallows a 2-kg fish that is at rest. If the larger fish swims at 1 m/s, what is its velocity immediately after lunch?

8.5 Conservation of momentum for inelastice collisions

m1v1 = v2

m1 + m2

Find the speed of the two fish after the inelastic collision