In a previous paper [1] two of the present authors considered
in detail the mechanics of a superball bouncing back and forth on
a rough horizontal plane. Reversals in direction in the horizontal
motion of the ball result from the application of a tangential law
of restitution at the point of impact of the ball and the plane. This
concept was first introduced by Garwin [2] who used a tangential
coefficient of restitution of one, which is not physically realistic.
Garwin’s model was modified by Cross [3] who employed a
tangential coefficient of restitution a satisfying 0oao1, with the
horizontal velocity of the point of impact of the ball being
reversed and reduced in magnitude by a factor of a in the impact.
Further details of the physics of this model are given in [1],
together with references to other theoretical and experimental
work.
All who have experimented with a superball will have at
sometime bounced the ball on the floor, followed by a bounce on a
vertical wall. If the bounce on the wall occurs while the ball is still
rising, it gives the ball some backspin, so that the direction of
motion is reversed at the next bounce on the floor resulting in the
ball hitting the wall a second time. With practice, the ball can be
made to bounce between the floor and wall several times. Such
motion is illustrated in the animations in Figs. 2, 3, 5, and 11. It is
our purpose to give a theoretical investigation of such motions
and the non-linear mappings which they engender. To this end we
establish in Section 2 the basic equations governing the model.
Essentially, each journey of the ball from floor to wall to floor,
assumed to take place in the same vertical plane, comprises four
events: (i) after launch from the floor the ball pursues a parabolic
trajectory until it hits the wall, (ii) the rebound from the wall,
(iii) the parabolic trajectory of the return journey to the floor and
(iv) the impact with the floor which provides the launch data for
the next excursion of the ball. The result of this analysis is the
derivation of a non-linear mapping which relates the floor launch
data (linear and angular velocity components of the ball and
distance from the wall) to the same parameters after the next
bounce on the floor.
In Section 3 some numerical trajectories of the non-linear
mapping are computed and examples given of motions with
various numbers of floor to wall bounces. Also illustrated are the
parameter spaces of initial conditions required to produce various
numbers of bounces off the wall. In Section 4 a scaling invariance
is introduced which rewrites the non-linear map of Section 2 in
terms of suitable canonical coordinates. This results in a threedimensional
non-linear map, a reduction in dimension by one
from the original system.
Section 5 presents some numerical results for the regions of
initial conditions which will result in a given number of bounces
against the wall in the canonical variables, analogous to those of
Section 3 for the original variables. The next two sections of the
work analyse these numerical results in some detail, focussing on
the behaviour of the mapping on two planes which comprise
boundaries of the region of interest. The paper concludes by
proposing a number of further questions related to the problem.
Before continuing to our analysis of the problem we have just
described, we note that there are limitations to the model of the
bounce of the superball that we use. It is recognised that th