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# Bouncing a Superball

A highly elastic superball can show some surprising behavior. When thrown down between two vertical planes, it will, in many circumstances, bounce back to near its initial location after three bounces. This Demonstration lets you control gravity (on or off), the initial velocity (by moving the arrow), the initial spin, the coefficient of normal restitution (elasticity; 1 denotes perfect elasticity), and the friction coefficient of the walls (1 means that there is no slippage when the ball strikes a wall). The color of the text inside the ball indicates the direction of spin (yellow denotes zero spin). You can experiment with either zero gravity or full gravity (980 ).

### DETAILS

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
Reference
[1] B. T. Hefner, "The Kinematics of a Superball Bouncing between Two Vertical Surfaces," American Journal of Physics, 72(7), 2004 pp. 875–883.

### PERMANENT CITATION

(Macalester College)
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