The Effect of Adding Mass
to a Tennis Ball
by Dr. Howard Brody
A tennis ball, to be approved for play by the ITF must have a mass between 56 and 59.4 grams. Therefore, a perfectly legal tennis ball can have a mass several grams greater (or less) than another perfectly legal ball. In addition, in the course of play, the ball may have additional mass due to moisture, dirt or clay adhering to the ball. How this small variation in mass affects the way the ball plays is the subject of this note.
There are two primary ways in which mass changes the playing characteristics of a tennis ball. The ratio of ball mass to effective racket mass determines the inherent "power" or Apparent Coefficient of Restitution (ACOR) of a racket. The heavier the ball, the lower is the ACOR of the ball-racket system, so the racket will have less power. This means the ball will have less speed as it leaves the strings.
A tennis ball, in its trajectory through the air, slows down due to air resistance. This drag force on the ball produces an acceleration (actually a deceleration) that is proportional to the inverse of the ball mass. A heavier ball will lose less speed in the air and end up at a higher speed than a lighter ball when it crosses the opposite baseline.
Let us investigate these two phenomena separately and then combine them, since the effects are in the opposite direction and they tend to cancel each other out.
Figure 1 displays the ACOR of a ball racket interaction as a function of ball mass. The value of ACOR influences the ball speed off the strings, Vh, and can be determined using the formula:
Vh = ACOR x Vinc + (1 + ACOR) x VR
Where Vinc is the incident ball speed and VR is the racket head speed.
Figure 2 shows the speed of a ground stroke as it crosses the opposite baseline versus ball mass for a 65 mph initial ball speed and for a ball speed adjusted for the change in ACOR. Note that the air resistance effect alone (fixed ACOR) will change the ball speed by 10% for a change of 10 grams in mass. When the ACOR is taken into account, the variation in final ball speed is of order 3% for a 10 gram change in mass.
Figure 3 shows the speed of a serve as it crosses the opposite baseline versus ball mass for both a constant initial ball speed (120 mph) and a speed adjusted to take into account the change in the ACOR.
Figure 4 shows the time it takes a serve to go from the racket to the baseline as a function of ball mass. Note that when the ACOR is taken into account, a variation of 10 grams in ball mass will change the time the receiver has by only a millisecond or so.
Conclusion: The addition of a few grams of material to a ball will not appreciably change the way that the game plays.
Dr. Howard Brody is an emeritus professor of physics at the University of Pennsylvania, where he was interim varsity tennis coach for part of the 1991 season. He played varsity tennis and earned his bachelor's degree at MIT and his master's and doctoral degrees at Cal Tech. For many years his primary field of research was elementary particle physics and high energy nuclear physics, working at laboratories in the US and in Europe. Recently he has been investigating the physics of sports, particularly tennis. He has written many papers and articles on the subject, given numerous lectures and talks on tennis, and done several television programs explaining the science behind tennis, football, and baseball. Professor Brody is a member the International Tennis Federation Technical Commission, the USTA Sports Science Committee, science advisor to the Professional Tennis Registry, technical advisor to the United States Racquet Stringers Association, on the Editorial Board of the Journal of Sports Engineering, and on the technical advisory panel of Tennis Magazine.
Dr. Brody's book Tennis Science for Tennis Players was published by the University of Pennsylvania Press in 1987, and his book (in conjunction with Rod Cross and Crawford Lindsey) The Physics and Technology of Tennis was published in 2002. He and Vic Braden are featured in a video "The Science and Myths of Tennis." As one of the six principal authors who have written an NSF sponsored high school physics course (Active Physics), he was responsible for the chapters dealing with the physics of sports. He has received the USPTR Plagenhoef award for sports science in 1996 and the International Tennis Hall of Fame Educational Merit Award for the year 2000. He is also one of the principal authors of the CD-ROM on The Science of Tennis funded by the Lawn Tennis Association Coach Education. At the 2003 Tennis Science and Technology Congress, the ITF in conjunction with Cislunar gave a prize for the best paper presented and named it the "Howard Brody Award" for his service to tennis.
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