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What is the Coefficient of Restitution?

By D. Poupon
Updated May 21, 2024
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The Coefficient Of Restitution (COR) measures the elasticity of a pair of objects in collision and is the ratio of their final relative speed to their initial relative speed. Determined through experimentation, this coefficient essentially measures bounciness. If the coefficient is close to one, then the collision is elastic and bouncy, if it is close to zero then collision is more plastic. Most professional sports have specific guidelines for the coefficient of restitution of equipment, such as balls, clubs and rackets.

Otherwise known as Newton’s Law of impact, the coefficient of restitution is determined through experimentation. The COR formula for objects that collide is defined as e = (v2-v1)/(u1-u2), where u1, u2 are the velocities before collision and v1, v2 are the velocities after collision of each object respectively. To determine the COR of an object dropped onto a stationary object such as the floor or a table the formula is e = √(bounce height / drop height). Velocities of objects moving horizontally or obliquely may be difficult to measure.

The COR can also be thought of as the proportion of kinetic energy retained by two objects before and after collision. If the coefficient is close to one, the collision is very elastic, like a table tennis ball dropped on a stainless steel table which has a COR of .92. Most of the kinetic energy present before the collision remains as kinetic energy after the collision. If the coefficient is close to zero, then collision is more plastic such as a piece of clay dropped on a wooden table; the clay sticks without bouncing at all. The kinetic energy has been converted into sound and other forms of energy, deforming the clay.

The bounciness of balls is a very important concept in sports, since materials with more spring could give an athlete an advantage. Since 1998, the United States Golfing Association had limited the coefficient of restitution of golf clubs with standard golf balls to .83. The COR of a National Basketball Association synthetic ball and a hard linoleum floor is .79; however, temperature, air pressure and other factors may impact this number.

In addition to giving players an unfair advantage, sports equipment that has unusually high coefficients of restitution may be unsafe. For instance composite bats are bouncier than wooden bats when struck by a baseball. Balls hit with composite bats may have unexpectedly high velocities and result in player injuries. In May 2009, the National Collegiate Athletic Association adopted the Bat-Ball Coefficient of Restitution (BBCOR) as a means of better testing baseball bat performance. All bats must have a BBCOR of less than 0.500.

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