# Dynamics of a Bouncing Ball

**Topics:**Measurement, Coefficient of restitution, Balls

**Pages:**9 (2746 words)

**Published:**August 28, 2012

Research Question: Of plywood, glass, stainless steel, and a ceramic tile, which least affects the dynamics of a bouncing table tennis ball?

Background Information: Table tennis is a ball game that can be played on any reasonably sized, flat, elevated surface. As is the case in any ball game, a crucial criterion to base which material to be used as a playing surface is the bounce of the ball. For any ball game to be fair to both sides, the playing surface must be such that the ball bounces back to a height that is as close as possible to the original height. Moreover, it is desired that the bounce be predictable. No ball game would be fun if the bounce is too uneven as this prevents the timing and strategic thinking of the player.

Theory and Explanation: The coefficient of restitution of an object is a fractional value representing the ratio of velocities before and after an impact. The coefficient of restitution can be calculated using the following equation: Cr = hH

Where:

* ‘Cr‘ is the coefficient of restitution,

* ‘h’ is the rebound height of the ball,

* ‘H’ is the height that the ball was initially dropped from.

This shows that the coefficient of restitution of a collision is the ratio of the square root of the rebound height (h) by the drop height (H). The coefficient of restitution varies from object to object and also according to the material which the object is impacted on (in this case bounced on). The value for the coefficient of restitution always ranges from 0 to 1 because ‘h’ cannot be greater than ‘H’. If a collision is perfectly elastic, the coefficient of restitution will be 1 and if the collision is perfectly inelastic, it will be 0. However, it is not practically possible for a collision to be perfectly elastic as in a collision energy is lost as it is converted to sound, heat and to overcome friction, and air resistance. The higher the coefficient of restitution of a collision, the smaller the difference between the bounce height and the drop height is. Since the purpose of this experiment is to determine which of the above materials least affect the dynamics of a bouncing ball, calculating for the coefficient of restitution of a bouncing table tennis ball on different surfaces is a perfect way of fulfilling this purpose. The material with the highest coefficient of restitution least affects the dynamics of a bouncing table tennis ball and vice versa.

By re-arranging the above equation as shown below, the equation appears in a standard linear equation y = mx + c, where ‘m’ is the gradient and ‘c’ is the y-intercept. hH=(Cr)2

h=H(Cr)2

h=HCr

In this equation, y is h, x is H and m is Cr. The y-intercept (c) is 0. By comparing different coefficient of restitution values, we can deduce which material least affects the dynamics of a bouncing ball.

Variables and their Manipulation:

Independent Variables:

* Drop height

* Material used

Dependent Variables:

* Bounce Height

* Coefficient of restitution

By controlling and changing the drop height, different values of the bounce height can be obtained in order for the coefficient of restitution for the particular material to be calculated. By using different materials as a rebound surface of the ball the coefficient of restitution of all these materials can be calculated and so the material which least affects the dynamics of a bouncing ball can be identified.

Experimental Constants:

The acceleration due to gravity for these experiments can be considered as constant as all the experiments are carried out in the same location and under the same conditions. The effects of air-resistance can also be ignored as the experiments are conducted in an air-conditioned laboratory. The table tennis ball used is kept constant for each one of the experiments as using a different ball may result in a different value for the coefficient of restitution The ruler used for measurements of the...

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