  We consider the vertical motion of a free falling ball bouncing elastically
on a racket moving in the vertical direction according to a regular
$1$-periodic function $f$. For fixed coprime $p,q$ we study existence,
stability in the sense of Lyapunov and multiplicity of $p$ periodic motions
making $q$ bounces in a period. If $f$ is real analytic we prove that one
periodic motion is unstable and give some information on the set of these
motions.
