# Coefficients of Restitution

## Abstract

The purpose of this experiment is to determine the coefficient of restitution for various balls.

## Introduction

The coefficient of restitution is the ratio of speeds of a falling object, from when it hits a given surface to when it leaves the surface. In laymen's terms, the coefficient of restitution is a measure of bounciness. A ball is a round or spherical object that is used most often in sports and games. Balls are made from different materials, but leather, rubber, and synthetics are the most common in modern times.

Balls have been a major part of almost all civilizations history. Some form of ball game is portrayed on early Egyptian monuments. Each spring two large groups of people, each representing one of their gods, acted out a contest that used a round, wooden ball and crooked sticks. The object was to drive the ball through the opposing goal which is the basis for almost all of today's modern ball games. Even among the Romans, who disliked participatory sports, ball play was extremely popular. The Roman baths set aside apartments for ball play, and many gentlemen had ball courts in their private villas. The ancient Roman ball was usually made of leather strips sewn together and filled with various materials, including animals.

## Diagram

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## Procedure

- We performed this experiment in Midwood High School, on the second floor, on an concrete surface.
- Take the ball and hold it at a set height above the surface. (We chose a height of 92 cm for all trials.)
- Drop the ball and record how high it bounces.
- Repeat for 5 trials.
- Repeat with various balls.
- Practice golf ball
- Wilson tennis ball
- rubber band ball (many rubber bands put together in ball form)
- Red plastic ball
- Generic unpainted billiard ball
- Rubber blue ball
- Painted wood ball
- Steel ball bearing
- Glass marble

## Analysis

The coefficient of restitution is found by the formula

Coefficient of Restitution = speed up/speed down.

In order to find speed we had to use the average height, that we measured, and put it in the formula

v = √(2gh)

Where v = velocity, g = 9.8m/s^{2}, and h = average height measured.

We took the average of the bounced height value (h) and put it in the formula along with the initial height (H) of 92 cm.

Coefficient of Restitution = √(2gh))/√(2gH) = √(h/H)

The results were then compiled in the table below (xls copy).

object | H (cm) | h_{1} (cm) |
h_{2} (cm) |
h_{3} (cm) |
h_{4} (cm) |
h_{5} (cm) |
h_{ave} (cm) |
c.o.r. |
---|---|---|---|---|---|---|---|---|

range golf ball | 92 | 67 | 66 | 68 | 68 | 70 | 67.8 | 0.858 |

tennis ball | 92 | 47 | 46 | 45 | 48 | 47 | 46.6 | 0.712 |

billiard ball | 92 | 60 | 55 | 61 | 59 | 62 | 59.4 | 0.804 |

hand ball | 92 | 51 | 51 | 52 | 53 | 53 | 52.0 | 0.752 |

wooden ball | 92 | 31 | 38 | 36 | 32 | 30 | 33.4 | 0.603 |

steel ball bearing | 92 | 32 | 33 | 34 | 32 | 33 | 32.8 | 0.597 |

glass marble | 92 | 37 | 40 | 43 | 39 | 40 | 39.8 | 0.658 |

ball of rubber bands | 92 | 62 | 63 | 64 | 62 | 64 | 63.0 | 0.828 |

hollow, hard plastic ball | 92 | 47 | 44 | 43 | 42 | 42 | 43.6 | 0.688 |

## Sources of Error

- The ball didn't bounce straight up because of uneven or cracked ground, which would have stopped the ball from reaching its peak height.
- When releasing the ball from your hand the ball rotates slightly, causing the ball to lose some of its translational energy which in turn causes the ball to not bounce as high as it would in a perfect world.

Jamin Bennett, Ruwan Meepagala -- 2006

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