Elastic and Inelastic Collisions

Summary

The following image shows an unfortunate situation: a van fails to stop as it approaches a line of traffic and hits a stationary car; and they move forwards together - is this elastic or inelastic collision?

Elastic and inelastic collisions

Without doing any calculations, we know that this is an inelastic collision because the crash transfers kinetic energy to other forms - the car is dented, so work must be done to deform the metal, and there is noise

However, we can calculate the kinetic energy transferred as shown in the example below...

In the world of atomic particles, collisions can be elastic because, for example, electrostatic charges can repel two atoms or nuclei without a transfer of kinetic energy to other forms

In the example above, the momentum before the crash is equal to the momentum after the crash:

Elastic and inelastic collisions

Example

Elastic and inelastic collisions

So the total kinetic energy is conserved in this collision. It is a useful rule to know that in an elastic collision the relative speeds of the two particles is the same before and after the collision.

More advanced problems in momentum

The generalised form of Newton's second law of motion

Elastic and inelastic collisions

Elastic and inelastic collisions

Collision in Two Dimensions

An interesting special result occurs when two atomic particles of the same mass collide elastically, when one of the particles is initially stationary

Elastic and inelastic collisions

If the two particles (protons for example) collide head-on, then the momentum and kinetic energy of the moving proton (A) is transferred completely to the stationary proton (B)

In this way both momentum and kinetic energy are conserved. This only happens when the particles are of the same mass; in all other cases both particles will be moving after the collision

What happens when two particles of the same mass collide, but the collision is not head on (as seen below)?

Elastic and inelastic collisions

The momentum of particle A can be resolved into two components: p cosθ along the line of collision and p sinθ perpendicular to the line of the collision

As in the first example, all of the momentum of A along the line of the collision (here p cosθ) is transferred to particle B

This leaves particle A with the component p sinθ, which is at right angles to the line of the collision

  • So, in a non-head-on elastic collision between two particles of the same mass, they always move at right angles to each other

And as you can see below, momentum is conserved is a vector quantity: p = pcosθ + psinθ

Kinetic energy is also conserved

  • Before the collision the kinetic energy is:

Elastic and inelastic collisions