Momentum and Energy in Collisions. A 2kg car moving at 10m/s strikes a 2kg car at rest. They stick together and move to the right at ___________m/s.

Momentum and Energy in Collisions

Starter

A 2kg car moving at 10m/s strikes a 2kg car at rest. They stick together and move to the right at ___________m/s.

Definition of Momentum

The symbol p stands for momentum.

Momentum is the product of mass and velocity.

p = mv

Examples of calculating momentum

A 2000kg car is moving at 30m/s. What isthe momentum of the car?

p = mv = (2000kg)(30 m/s) = 60,000 kg m/s

A .1 kg bullet has a momentum of 50 kg m/s. How fast is it moving?

v = p/m = 50/.1 = 500 m/s

More Examples:

A 10kg rock and a 2 kg rock have the same momentum of 100 kg m/s. What is the speed of each rock?

Answer: For the 10kg rock: 100 = 10v, 0r v = 10m/s.

For the 2 kg rock, 100 = 2v, or v = 50 m/s.

The Vector Nature of Momentum

Momentum is a vector – it points in the same

direction as the velocity.

In one dimension, momentum pointing to the right is positive.

Momentum pointing to the left is negative.

Example:

Find the momentum of each ball. Be careful of the signs!

Answer: For the 3kg ball, p = 3(20) = 60 kg m/s

For the 10 kg ball, p = 2(-10) = -20 kg m/s

Newton’s 2nd Law in terms of Momentum

Favg = maavg = mDv/Dt = = Dp/Dt

Favg Dt = Dp

Impulse = Change in momentum

Impulse

Area = Favg Dt = Impulse = Dp

Example

This force is applied to a 3kg particle moving at 4m/s.

1. What is the impulse?Impulse = area = ½bh = ½(3)(3) =4.5Ns

2. How fast is the particle moving after 4 seconds?

Impulse = Dp I = mvf – mvi 4.5 = 3vf – 3(4) vf = 5.5 m/s

Conservation of Momentum

Momentum is Conserved for Collisions

Total momentum = Total momentum before the collision after the

collision

Pbefore = Pafter

Types of Collisions

• Elastic ( Energy and Momentum are conserved)

• Inelastic ( Only momentum is conserved)

• Note: Momentum is Always Conserved for any collision.

Completely Inelastic Collisions

• When two objects hit and stick together.• Or, the reverse of this – when one object breaks apart into two objects.

Momentum is Conserved

Total momentum = Total momentum before the collision after the collision

Pbefore = Pafter

ExampleA cannon ( mass = 500kg) fires a cannon ball ( m = 50kg) at 40m/s.How fast does the cannon move after it fires the cannon ball?

Before: Pi = 0

After: Pf = mballvball + mcannonvcannonPi = Pf

(-mballvball )/mcannon = vcannon = (-50)(40)/500 = -4 m/s

0 = mballvball + mcannonvcannon

Example

A car mass = 1kg moving at 3m/s hits another 1kg car and they stick together.How fast are they moving after they stick together?

Pi = mvi = 1(3) = 3 Pf = 2mv = 2v 2v = 3, v = 1.5 m/s

Example

A car mass = 10kg moving at 2m/s hits another 15kg car moving to the left at3m/s and they stick together. How fast are they moving after they stick together?

Pi = m1v1i + m2v2i = 10(2) + 15(-3) = -25

Pf = m1v1f + m2v2f = (m1 + m2 )vf = 25vf

-25 = 25vf vf = -25/25 = -1 m/s

Starter

A 2kg car moving at 24m/s strikes a 10kg car at rest. They stick together and move to the right at ___________m/s.

M

M

2 particle,1-D Elastic Collisions

Momentum is conserved: m1v1i + m2v2i = m1v1f + m2v2f

Energy is Conserved :

v1i + v1f = v2i + v2f

This gives you 2 equations and 2 unknowns.

Example

A 10kg ball moving to the right at 3m/s strikes a 5kg ball at rest.

Find the velocity of each ball after the collision.

m1v1i + m2v2i = m1v1f + m2v2f

10(3) + 0 = 10v1f + 5v2f

OR (1) 30 = 10v1f + 5v2f

v1i + v1f = v2i + v2f OR (2) 3 + v1f = v2f

CONTINUED………

(2) 3 + v1f = v2f

(1) 30 = 10v1f + 5v2f

The problem now is to solve two equations and two unknowns.

Sub (2) into (1) : (1) 30 = 10v1f + 5 ( 3 + v1f )

30 = 10v1f + 15 + 5v1f , 15 = 15v1f , v1f = 1 m/s

Then (2) : 3 + v1i = v2f , 3 + 1 = 4 = v2f

EXIT

If an elastic collision takes place between two identical objects, they undergo velocity exchange.

A 2kg ball moving to the right at 5m/s collides with a 2kg ball moving to the leftat 3m/s. What are their final velocities?