Snapshot 1: the ball bounces up and out of the container when the settings are realistic: gravity is Earth's gravity, elasticity is the superball elasticity of 0.85, and there is no slippage along the wall

Snapshot 2: with zero gravity, a ball with only 0.65 elasticity can still end up higher than its starting point

Snapshot 3: uses the same settings as Snapshot 1, but with an initial spin, which affects the motion

The way that the parameters affect what happens at each bounce is described in detail in [1]. The main idea is that the laws of conservation of angular momentum and energy give equations that yield, at each bounce, new values of spin and velocity from the old values, with suitable modifications when either elasticity or friction is less than its ideal value. Between bounces the ball continues on its path, with a vertical acceleration if gravity is present.

To describe what happens at each bounce, use the following notation. Let

, where

is the angular velocity and

is the velocity of the center, representing the state of the ball's velocity. Let

be the elasticity and let

be the tangential restitution (which ranges from

to

and is

, where

is the coefficient of friction of the walls). In the ball's moment of inertia

, let

. If

and

are used to denote the state just before and just after the bounce, then the central transformation is elegantly given in matrix form as

[1] B. T. Hefner, "The Kinematics of a Superball Bouncing between Two Vertical Surfaces,"

*American Journal of Physics,* **72**(7), 2004 pp. 875–883.