Measurements of the horizontal coefficient of restitution for a superball and a tennis ball

Abstract

When a ball is incident obliquely on a flat surface, the rebound spin, speed, and angle generally differ from the corresponding incident values. Measurements of all three quantities were made using a digital video camera to film the bounce of a tennis ball incident with zero spin at various angles on several different surfaces. The maximum spin rate of a spherical ball is determined by the condition that the ball commences to roll at the end of the impact. Under some conditions, the ball was found to spin faster than this limit. This result can be explained if the ball or the surface stores energy elastically due to deformation in a direction parallel to the surface. The latter effect was investigated by comparing the bounce of a tennis ball with that of a superball. Ideally, the coefficient of restitution (COR) of a superball is 1.0 in both the vertical and horizontal directions. The COR for the superball studied was found to be 0.76 in the horizontal direction, and the corresponding COR for a tennis ball was found to vary from −0.51 to +0.24 depending on the incident angle and the coefficient of sliding friction.

The dynamics of a bouncing superball with spin

“…The corresponding trajectory of the ball can be followed backwards in time until the ball reaches the floor, which will give the quantities immediately after a non-existent zeroth bounce. These constitute the initial conditions for the iterative relations (1), (4) and (5). Since the horizontal and angular velocities are constant during the flight, then clearly u 0 =ũ, ω 0 =ω, and v 0 follows fromṽ andỹ.…”

“…A model of the bounce of a superball was described by Cross [5]. This model was extended by Hefner [10] to include the parabolic trajectory between bounces, and is briefly reviewed here.…”

“…Doménech [7] extended the model of Garwin and Cross to three dimensions and also obtained consistent experimental results giving the horizontal coefficient of restitution as 0.5. In this paper, we consider the bouncing behaviour of a superball with spin on a flat surface in more detail, using the model for the bounce described by Cross [5]. In particular, conditions on the initial conditions for a reversal of direction or spin of the ball are considered.…”

“…Cross [5] compared the bounce of a superball with the bounce of a tennis ball whereas Garwin [9] was particularly interested in the phenomenon where the superball reverses direction when it is thrown forward to bounce between two horizontal parallel surfaces. This behaviour was also investigated by Hefner [10] but in the case where the parallel surfaces are vertical.…”

“…To model the bounce of a superball, the first of these two concepts was extended by Garwin [9] who considered a similar model in the horizontal direction wherein the horizontal motion of the ball at the point of impact is reversed at the bounce. Cross [5] modified Garwin's model by introducing the second component of a horizontal coefficient of restitution that reduces the horizontal velocity of the ball after the bounce by a fixed amount. In both cases, this equation describing the change in the horizontal velocity at the bounce was combined with an equation representing the conservation of angular momentum.…”