Tennis Balls and Racquet Speeds

If the top of a tennis racket reaches 100 mph at the point when it strikes the ball, at what speed should one expect the ball to reach when it leaves the racket? It is understood that many variables determine this speed but we are interested in the relative speed between the speed of the racket and the speed of the ball.
- Mike (age 63)
St Petersburg Florida USA

I’m going to make a guess of about 63 mph.  My reasoning is that the official USTA specification for the bounce of  tennis balls is about 55%, i.e. they require that a tennis ball dropped from 100 inches rebound to a height of 55 inches plus or minus 2. The coefficient of restitution (COR), i.e. the relative before and after velocities, is equal to the square root of the height ratio, which gives 0.74. Now turn your problem around:  suppose you hold a tennis racquet rigidly and someone throws a tennis ball against it at a speed of 100 mph.  All things being equal it should rebound at a speed of 74 plus or minus a couple of mph.   However, there is another factor involved.  There is an additional coefficient involved that is due to the fact that the tennis ball interacting with a stone floor at the USTA test site is not the same as when the ball interacts with a tennis racquet. Howard Brody, a tennis enthusiast at the University of Pennsylvania, suggests that COR of the ball-racquet impact is about 0.85.  So multiplying the two CORs together leads me to the answer of 63 mph.  I would be happy if my answer were within 10% of the actual value.

Happy serving,
LeeH 

(published on 02/17/2008)