A Billiard-Theoretic Approach to Elementary 1d Elastic Collisions

Abstract

A simple relation is developed between elastic collisions of freely-moving point particles in one dimension and a corresponding billiard system. For two particles with masses m_1 and m_2 on the half-line x>0 that approach an elastic barrier at x=0, the corresponding billiard system is an infinite wedge. The collision history of the two particles can be easily inferred from the corresponding billiard trajectory. This connection nicely explains the classic demonstrations of the ``dime on the superball'' and the ``baseball on the basketball'' that are a staple in elementary physics courses. It is also shown that three elastic particles on an infinite line and three particles on a finite ring correspond, respectively, to the motion of a billiard ball in an infinite wedge and on on a triangular billiard table. It is shown how to determine the angles of these two sets in terms of the particle masses.Comment: 7 pages, 8 figures, 2-column revtex4 format, for submission to the American Journal of Physics. Introductory text, several references and one figure added in response to referee comments. A few more additions and corrections; final version for AJP. Yet a few more changes and one minor error corrected just before publicatio