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### Bouncing Along with Mathematics in Sport

August25

My Dad was a keen sportsman and encouraged me to play cricket in the summer. I can still remember the thrill of getting my first cricket bat and playing having Dad show me how to hold the bat and play the hard cricket ball.

Almost every type of ball used in a sporting event must bounce according to the rules of the game. If the ball bounces too high or too low, the players will complain that something is wrong with it.

The standard test for bounce is to drop a ball from a certain height onto a hard surface such as a slab of concrete and then measure how high it bounces. When a tennis ball is dropped from a height of 100 inches (2.54 m) it must bounce to a height between 53 inches (1.35 m) and 58 inches (1.47 m). For official use in major tournaments, tennis balls must be properly tested and approved, for a moderately large fee, but the fee is only a small fraction of the total value of balls sold. Tennis courts themselves vary in hardness, which affects bounce height, so a standard surface such as concrete is used for these tests.

There is no such official rule for a cricket ball. There is simply a tradition that is monitored by umpires, and one that is an industry standard. When a cricket ball is dropped from a height of 2.0 m onto a heavy steel plate, it bounces to a height somewhere between 0.56 m and 0.76 m. Cricket balls are a lot less bouncy that tennis balls and the permitted range of possible bounce heights is larger. A useful way of specifying the bounce is to take the ratio of the bounce speed to the incident speed. When a ball is dropped from a height of 2.0 m it lands at a speed of 6.26 m/s, regardless of the type or weight of the ball. A cricket ball bounces to about one third of that height (0.67 m), in which case it rebounds at a speed of 3.61 m/s. The ratio of these two speeds is 3.61/6.26 = 0.58 and is called the coefficient of restitution (COR). The COR of a tennis ball is about 0.75. The COR determines not only the bounce height but also the speed at which a ball comes off the bat. The batted ball speed also depends on the speed of the bat. Article source here.

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