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.

Definition of Momentum The symbol p stands for momentum. Momentum is the product of mass and velocity. p = mv

A 2000kg car is moving at 30m/s. What is the 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

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.

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

F avg = ma avg = m  v/  t = =  p/  t F avg  t =  p Impulse = Change in momentum

Impulse Area = F avg  t = Impulse =  p

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 =  p I = mv f – mv i 4.5 = 3v f – 3(4) v f = 5.5 m/s

Momentum is Conserved for Collisions Total momentum = Total momentum before the collision after the collision P before = P after

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 P before = P after

Example A 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: P i = 0 After: P f = m ball v ball + m cannon v cannon P i = P f (-m ball v ball )/m cannon = v cannon = (-50)(40)/500 = -4 m/s 0 = m ball v ball + m cannon v cannon

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? P i = mv i = 1(3) = 3 P f = 2mv = 2v 2v = 3, v = 1.5 m/s

A car mass = 10kg moving at 2m/s hits another 15kg car moving to the left at 3m/s and they stick together. How fast are they moving after they stick together? P i = m 1 v 1i + m 2 v 2i = 10(2) + 15(-3) = -25 P f = m 1 v 1f + m 2 v 2f = (m 1 + m 2 )v f = 25v f -25 = 25v f v f = -25/25 = -1 m/s

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

Momentum is conserved: m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f Energy is Conserved : v 1i + v 1f = v 2i + v 2f 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. m 1 v 1i + m 2 v 2i = m 1 v 1f + m 2 v 2f 10(3) + 0 = 10v 1f + 5v 2f OR (1) 30 = 10v 1f + 5v 2f v 1i + v 1f = v 2i + v 2f OR (2) 3 + v 1f = v 2f

CONTINUED……… (2) 3 + v 1f = v 2f ( 1) 30 = 10v 1f + 5v 2f The problem now is to solve two equations and two unknowns. Sub (2) into (1) : (1) 30 = 10v 1f + 5 ( 3 + v 1f ) 30 = 10v 1f v 1f, 15 = 15v 1f, v 1f = 1 m/s Then (2) : 3 + v 1i = v 2f, = 4 = v 2f

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 left at 3m/s. What are their final velocities?