MadSci Network: Physics

Re: When you line up a bunch of balls, why does the top one bounce high?

Date: Wed Feb 9 08:06:12 2000
Posted By: Georg Hager, Grad student, Theoretical Particle Physics
Area of science: Physics
ID: 948757975.Ph

Dear Greg!

You can be sure of one thing: The top one will definitely not bounce into space. This is a simple matter of energy: Even if you could manage to somehow `pump' all the energy of the other eight balls into the top one, it would simply bounce nine times higher as the height from which it was dropped. This is a consequence of energy conservation: The sum of potential energy and kinetic energy in the system cannot change (if there is no dissipation, like friction etc.).

The interesting question is now, what is possible to achieve using such a setup? How high can the top ball possibly bounce? We have to distinguish two cases here:

  1. The balls are disconnected. Imagine there being a small space between each ball and the next when they are dropped. This way we can assume that each ball-ball collision takes place independently from the others. All you have to know here is that if two objects with the same mass collide with the same velocity, each one just reverses its direction of motion. It is then easy to see that at the end (after the bounce, when all balls move upward again) the balls all have the same velocity, and will thus bounce to the height from which they were dropped, each seperately. So there is no gain here.
  2. The lower balls (all of them) are connected to each other, effectively `acting' as a single ball with eight times the mass. The top ball is assumed to be disconnected, like in case 1. What we have to study here is the head-collision of two balls with equal speed but different mass. Physics can handle this using energy and momentum conservation. Let me just tell you here that in the case where the heavier ball is much heavier than the top one (eight times the mass will sort of qualifyfor that), the top ball will fly away with twice the velocity it had before the collision, thereby reaching four times its previous height. You can get this result by placing yourself onto (physicists say `into the reference frame') of the heavy ball. The collision with the light ball will hardly influence the heavy one, and the light one will just reverse its directon. Looking at the situation after the collision `from the ground' you see that the light ball has now twice the velocity it had before.
Hope that helps,

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