  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.
