A particle of mass bounces back and forth between two walls of mass moving towards each other in one dimension. The walls are separated by a distance . The wall on the left has velocity and the wall on the right has velocity . The particle has speed . Friction is negligible and the particle-wall collisions are elastic.
Consider a collision between the particle and the wall on the right. Show that the centre-of-mass velocity of the particle-wall system is . Calculate the particle's speed following the collision.
Assume that the particle is much lighter than the walls, i.e., . Show that the particle's speed increases by approximately every time it collides with a wall.
Assume also that (so that particle-wall collisions are frequent) and that the velocities of the two walls remain nearly equal and opposite. Show that in a time interval , over which the change in is negligible, the wall separation changes by . Show that the number of particle-wall collisions during is approximately and that the particle's speed increases by during this time interval.
Hence show that under the given conditions the particle speed is approximately proportional to .