Physics 1D03 - Lecture 26

Serway & Jewett 9.3 - 9.5

Collisions and Momentum

• Conservation of Momentum • Elastic and inelastic collisions

Physics 1D03 - Lecture 26

Collisions

A collision is a brief interaction between two (or more) objects. We use the word “collision” when the interaction time Δt is short relative to the rest of the motion.

During a collision, the objects exert equal and opposite forces on each other. We assume these “internal” forces are much larger than any external forces on the system.

We can ignore external forces if we compare velocities just before and just after the collision, and if the interaction force is much larger than any external force.

m1 m2

v1,i v2,i

v1,f v2,f

F1

F2 = -F1

Physics 1D03 - Lecture 26

Elastic and Inelastic Collisions

Momentum is conserved in collisions. Kinetic energy is sometimes conserved; it depends on the nature of the interaction force.

A collision is called elastic if the total kinetic energy is the same before and after the collision. If the interaction force is conservative, a collision between particles will be elastic (eg: billiard balls).

If kinetic energy is lost (converted to other forms of energy), the collision is called inelastic (eg: tennis ball and a wall).

A completely inelastic collision is one in which the two colliding objects stick together after the collision (eg: alien slime and a spaceship). Kinetic energy is lost in this collision.

Physics 1D03 - Lecture 26

Quiz

In dense regions of galaxies, or when galaxies collide, the stars are said to collide. Even though they actually do not hit each other, what type of collision do they undergo:

a) elasticb) inelasticc) depends on whether they form a binary star or not

Physics 1D03 - Lecture 26

If there are no external forces, then the total momentum is conserved:

p1,i + p2,i = p1,f + p2,f

This is a vector equation. It applies to each component of p separately.

m1 m2

v1,i v2,i

v1,f v2,f

Physics 1D03 - Lecture 26

Elastic Collisions

In one dimension (all motion along the x-axis):

1) Momentum is conserved:

ffii 22112211 vmvmvmvm

2222

12112

12222

12112

1ffii

vmvmvmvm

In one dimension, the velocities are represented by positive or negative numbers to indicate direction.

2) Kinetic Energy is conserved:

We can solve for two variables if the other four are known.

Physics 1D03 - Lecture 26

One useful result: for elastic collisions, the magnitude of the relative velocity is the same before and after the collision:

|v1,i – v2,i | = |v1,f – v2,f |(This is true for elastic collisions in 2 and 3 dimensions as well).

An important case is a particle directed at a stationary target (v2,i = 0):

• Equal masses: If m1 = m2, then v1,f will be zero (1-D).• If m1 < m2, then the incident particle recoils in the opposite direction.• If m1 > m2, then both particles will move “forward” after the collision.

before after

Physics 1D03 - Lecture 26

Elastic collisions, stationary target (v2,i = 0):

Two limiting cases:1) If m1 << m2 , the incident particle

rebounds with nearly its original speed.

v1

-v1

2) If m1 >> m2 , the target particle moves away with (nearly) twice the original speed of the incident particle.

v1 v1 2v1

Physics 1D03 - Lecture 26

Quiz

A tennis ball is placed on top of a basketball and both are dropped.The basketball hits the ground at speed v0. What is the maximum speed at which the tennis ball can bounce upward from the basketball? (For “maximum” speed, assume the basketball is much more massive than the tennis ball, and both are elastic).

a) v0

b) 2v0

c) 3v0 ?

v0

v0

Physics 1D03 - Lecture 26

Example – inelastic collision:

A neutron, with mass m = 1 amu (atomic mass unit), travelling at speed v0, strikes a stationary deuterium nucleus (mass 2 amu), and sticks to it, forming a nucleus of tritium. What is the final speed of the tritium nucleus?

Physics 1D03 - Lecture 26

2 kg 4 kg

6 m/s 5 m/s

An elastic collision:

Two carts moving in opposite direction collide and bounce back. If cart 1 bounces back with v=2m/s, what is the speed of cart 2 ?

Physics 1D03 - Lecture 26

Example

A 1500kg car traveling east with a speed of 25m/s collides with a 2500kg car traveling north at a speed of 20m/s. Find the direction and velocity of the cars after the collision, assuming that they stick together.

Physics 1D03 - Lecture 26

Summary

• Momentum is conserved in collisions.• In elastic collisions, kinetic energy is also conserved.