Bouncing Balls and Geometric Progressions

Observing the bouncing of a marble on a table is a rather common experience. The tic-tac sound of the rigid ball, nevertheless, carries quite a pleasant surprise. In fact, by measuring the total time of bouncing [delta]t, the coefficient of restitution can be estimated. As is well known, in an inelastic collision the kinetic energy is not conserved, and therefore the speed decreases. The speeds v[subscript i] and v[subscript f], before and after the collision occurs, respectively, are related as follows:[nu][subscript f]=[epsilon][nu][subscript i], (1) where [epsilon] < 1. By measuring the initial height h[subscript 0] from which the marble is released, we find that [equation omitted], (2) where g is the acceleration due to gravity and T is the total time from initial release until the ball stops bouncing.