To determine the motion of the ball we must calculate the times, hence phases (from eq. (1.2)), when the ball and the table collide. An impact occurs when the difference between the ball position and the table position is zero. Between impacts, the ball goes up and down according to Newton's law for the motion of a projectile in a constant gravitational field of strength g. Since the motion between impacts is simple, we will present the motion of the ball in terms of an impact map, that is, some rule that takes as input the current values of the impact phase and impact velocity and then generates the next impact phase and impact velocity.
Let
be the ball's position at time t after the kth impact, where 
is the
position at the kth impact and 
is the time of the kth impact, and let
be the table's position with an amplitude A, angular frequency 
,
and phase 
at t = 0. We add one to the sine function
to ensure that the table's amplitude is always positive. The difference in
position between the ball and table is
which should always be a non-negative function since the
ball is never below the table. The first value at which d(t) = 0, 
,
implicitly defines the time of
the next impact. Substituting equations (1.8) and (1.9) into equation
(1.10) and setting d(t) to zero yields
Equation (1.11) can be rewritten in terms of the phase when
the identification 
is made between the phase
variable and the time variable. This leads to the implicit phase
map of the form,
where 
is the next 
for which 
. In
deriving equation (1.12) we used the fact
that the table position and the ball position are identical at an impact; that
is, 
.
An explicit velocity map is derived directly from the impact relation, equation (1.7), as
or, in the phase variable,
noting that the table's velocity is
just the time derivative of the table's position, 
,
and that, between impacts, the ball is subject to the acceleration of gravity,
so its velocity is given by
.
The overdot is Newton's original
notation denoting differentiation with respect to time.
The implicit phase map (eq. (1.12)) and the explicit velocity
map (eq. (1.14)) constitute the exact model for the bouncing
ball system. The dynamics of the bouncing ball are easy to simulate on a
computer using these two equations. Unfortunately, the phase map is an implicit
algebraic equation for the variable ; that is,
cannot be isolated from the other variables. To solve the
phase function for a numerical algorithm is needed to locate the
zeros of the phase function (see Appendix A). Still, this
presents little problem for numerical simulations, or even, as we shall see,
for a good deal of analytical work.
