The height to which a ball will bounce depends on the height from which
it is dropped, what the ball is made out of (and if it is inflated,
what the pressure is), and what the surface it bounces from is made out
of. The radius of the ball doesn't really matter, if you are measuring
the height of the ball from the bottom of the ball to the ground.
A ball's gravitational potential energy is proportional to its
height. At the bottom, just before the bounce, this energy is now all
in the form of kinetic energy. After the bounce, the ball and the
ground or floor have absorbed some of that energy and have become
warmer and have made a noise. This energy lost in the bounce is a more
or less constant fraction of the energy of the ball before the bounce.
As the ball goes back up, kinetic energy (now a bit less) gets traded
back for gravitational potential energy, and it will rise back to a
height that is the original height times (1-fraction of energy lost).
We'll call this number f. For a superball, f may be around 90% (0.9) or
perhaps even bigger. For a steel ball on a thick steel plate, f is
>0.95. For a properly inflated basketball, f is about 0.75. For a
squash ball, f might be less than 0.5 or 0.25 - squash balls are not
very bouncy. The steel ball on an unvarnished pine wood floor may not
bounce at all, but rather make a dent, and so what the floor is made
out of makes quite a lot of difference.
For multiple bounces, it's just like dropping the ball again from
a reduced height. If the first height is h, the second will be f*h, the
third f*f*h, the fourth f*f*f*h, and so on. So if f is 0.9, the first
bounce will be 0.9 times as high, the second 0.81 times as high, the
third 0.729 times as high (as the original height), and so on.
Try it yourself! Does f depend on the height? (it shouldn't much, but it might..) Try it for different balls!
(republished on 07/11/06)