  The dynamical properties of a particle in a gravitational field colliding
with a rigid wall moving with piecewise constant velocity are studied. The
linear nature of the wall's motion permits further analytical investigation
than is possible for the system's sinusoidal counterpart. We consider three
distinct approaches to modeling collisions: (i) elastic, (ii) inelastic with
constant restitution coefficient and (iii) inelastic with a velocity-dependent
restitution function. We confirm the existence of distinct unbounded orbits
(Fermi acceleration) in the elastic model, and investigate regular and chaotic
behavior in the inelastic cases. We also examine in the constant restitution
model trajectories wherein the particle experiences an infinite number of
collisions in a finite time i.e., the phenomenon of inelastic collapse. We
address these so-called "sticking solutions" and their relation to both the
overall dynamics and the phenomenon of self-reanimating chaos. Additionally, we
investigate the long-term behavior of the system as a function of both initial
conditions and parameter values. We find the non-smooth nature of the system
produces novel bifurcation phenomena not seen in the sinusoidal model,
including border-collision bifurcations. The analytical and numerical
investigations reveal that although our piecewise linear bouncer is a
simplified version of the sinusoidal model, the former not only captures
essential features of the latter but also exhibits behavior unique to the
discontinuous dynamics.
