There is more to tennis than meets the eye. Acing a serve or a backhand stroke does not just happen because of practice; it has very much to do with physics. The physics of tennis is often overlooked for the major role it plays in the sport. Currently, critics think that the actual sport of tennis is dwindling because it has become a game of speed rather than technique or talent. People are winning the game by acing a serve rather than a well-played out point. In this paper, we will address the tennis ball, and how it is changing to slow down the game, and the racket in terms of the sweet spot, the tension of the strings, and how the hold of the racket can affect the swing. The racket-ball relationship is demonstrably important to the physics of tennis too.

The tennis ball is a unique element of the game. A tennis ball’s trajectory is based on its speed, spin, height, density, diameter, drag constant, and, of course, the fuzz. At present, the International Tennis Federation is brainstorming ways to slow down the course of the tennis ball in order to elongate the game of tennis.

All of the forces must be added together to determine the magnitude of the force and the direction it will act. In a mathematical nutshell it is:

where F is the force and W is the mathematical notation for weight, L for Lift and D for Drag. In terms of slowing the ball down, it would make sense to enlarge its diameter. The plan is to increase a standard-size ball’s diameter between six and twenty percent. "This will slow the ball down in flight by aerodynamic drag and give the receiver a chance to return the serve," says The Daily Telegraph, who reported on the annual American Association for the Advancement of Science meeting in 1995. There is some hesitation about this method because as a ball increases its diameter, the drag coefficient lowers, so that the engineers achieve the opposite result. The drag coefficient is the ratio of the drag on a body moving through air to the product of the velocity and the surface area of the body.

where F represents the force of the drag, k is the constant, V equals the velocity and A is the cross-sectional area. Due to the fuzz on a tennis ball, however, the drag coefficient stays the same so that we can enlarge the ball through its diameter and it will still slow down the game. The drag constant does vary slightly as the fuzz on a ball decreases due to hard hitting, which is why professionals use new balls every nine games or so. The picture below shows a tennis ball’s path in terms of air resistance.

Surprisingly, there are no specific rules concerning the tennis ball. This means that the shape or the weight of the ball could be changed. By altering the property of the ball it could slow down a match without changing the game of tennis. Steve Haake, a consultant for the International Tennis Federation says,

"Balls that have either a higher mass or stiffness were found to rebound faster from the racket. This is because the racket strings deform more than they would with a lighter or less stiff ball. More energy will therefore be returned to the heavier or stiffer ball, causing it to rebound at a higher speed."

This source confirms that in order to create a more efficient tennis ball that will slow down the game of tennis, one must not only consider the diameter of the ball, but it’s physical form too.

The tennis racket also plays a key role in the game of tennis, as seen above. The size of the racket, and how one holds it, has an effect on the game. Tennis racket size had not changed in over half a century, because people thought that "any radical chance would feel wrong and require the player to relearn to some degree," until Head took a chance. Now Brody "fears that improvements in racket technology are making tennis too fast and consequently…it has become a serving game where you don’t see rallies."

So what has happened to the rackets? Head produced the Prince racket with an oversized head in keeping with "the same overall length, weight and balance as a conventional racket so that it would "feel" the same as a normal racket." Beyond that, the rackets are "lighter, stiffer and more powerful" than the wooden rackets used so many years ago. Howard Brody comments,

"the result is a racket which has a larger rebound coefficient [leading

to more power], can be swung faster [yielding more power], is more

maneuverable, and is more forgiving [you don’t have to hit the ball at

exactly the center of the head or any other sweet spot]."

At present, the International Tennis Federation is attempting to create some rules for the racket as a possible solution for the increased speed in the game. The way in which one holds a racket is affected by the style of the actual racket. Professional players have changed their strokes from a "classic stand sideways, long swing, firm wrist, move weight forward style to an open stance, semi-western grip, and really whip the racket around because the new rackets let them get away with this." Therefore technique is also dwindling because of this new style in racket.

Brody experimented with rackets suspended in air in order to achieve concrete facts about handling rackets. If a racket is hit at the center of mass, then the racket will respond, but not rotate. If a ball hits at a certain distance from the center of mass, then the racket will respond and also rotate. There is a specific part of the handle that will not move when the ball hits this certain distance, allowing the force of the hand to be used to direct the ball rather than wasting it on just holding the racket. When holding a racket, one must overcome the velocity of the ball and the tendency to rotate if the ball does not hit this perfect spot, otherwise known as the "sweet spot," which will take away from the actual shot. To be perfectly clear, the sweet spot is the distance between the handle spot and the area away from the center of mass where the ball hits that is ideal. The result is that the player will get the most power out of his shot.

The physics of tennis is a complex issue. We discovered, though our research, that there are many components to the physics of tennis. We also discovered that many issues lie in the development of the ball. In order to preserve the game, physicists are not only researching the tennis ball but also the racket ball relationship. We learned much about both of these topics through extensive reading and observation.

In recent years, many fans and spectators have complained that the game of tennis has become too fast, and too predictable. Points with long rallies have been cut down due to new racket and other technologies. Tennis has turned into a serve and return game, where the point ends after two shots. One way to remedy this problem is to change certain properties of the tennis ball. We propose to experiment this theory by testing different ball sizes and fuzziness (friction). We will model our experiments after a professional player’s average serve velocity by keeping this velocity constant throughout the procedure. We plan to use Interactive Physics to conduct the experiment. In Interactive Physics we can change variables easily to achieve success in our experiment.

We had to do some research about ball regulations and other rules before beginning the experiment. We looked on the United States Tennis Association website (www.usta.com) and found exactly what we were looking for. The rules require the mass of the tennis ball to be between 56 and 59.4 grams, and the bound of the ball must be between 134.6 and 147.32 centimeters when dropped 254 cm. This will determine the elasticity of the ball. Elasticity is the property of returning to an initial form or state following deformation. We learned that there is no set amount of fuzz nor fuzz on the ball. These guidelines will give us some area to play with when we are conducting our experiment.

There are many steps involved in using Interactive Physics. First you have to set up the experiment scene, which includes a ball, an anchored base, and a set velocity for the ball that imitates a pro server. A professional server usually serves the ball at 100 miles per hour, or 45 meters per second. By double clicking on the ball you have made, a window will appear, which shows all of the properties of the ball. Here, you can set the velocity, and the mass. At the bottom of the screen, you can alter the radius and under "" you can set the air resistance, or in this case, drag coefficient. By clicking once on the ball, a blue arrow will appear, which you can drag out to increase the velocity or move it around 360 degrees to change the direction. We will set the velocity to 45 meters per second and then pick numbers for the elasticity and air resistance. The ball will be launched at 2 meters above the ground, which is the average height of a tennis player and raquet. After setting up the ball and velocity the way you want it, you can click "run" at the top of the screen to stimulate the experiment.

We will do the experiment ten times, with different radii of the tennis ball, since this is the proposed method of saving the game of tennis. Starting with .0635 meters, and then .0638m, .0641m, .0644m, .0647m, .065m, .0653m, .0657m, .0661m, and .0666m. Then we will make a chart with ten rows and six columns as shown below.

trial	initial V	radius	first drop	10cm later	time	final V
1
2
3
4
5
6
7
8
9
10

This chart will assist us in collecting our data.

From our previous research, we conclude that the bigger the radius of the ball will lead to more friction and slower final velocity, which will allow the game of tennis to proceed longer, because the receiver will have a better chance to return it. The physics of tennis is a complex issue, however. The racket and racket-ball relationship must also be taken into perspective when considering the game of tennis. For our purposes, we will only consider the dynamics of the tennis ball, and how it affects the game.

The primary focus of our research on tennis is in how to slow down the game. The current issue we are faced with is that tennis has become too fast a game, which creates frustration for players and spectators. The solution we experimented with was enlarging the diameter of the tennis ball, in an attempt to slow down the pace of the game. The idea behind changing the size of the tennis ball is that the final velocity of the ball (for our purpose, measured 10 cm after the first bounce) would be slower, enabling the receiver to return easier, hence allowing to longer rallies and tennis games. We used Interactive Physics, creating a simulated tennis scenario in order to test out this theory.

The initial velocity of the tennis ball was set to 45 m/s, the average speed of a professional serve, and the mass was set to .056 kg, that of a tennis ball. There is no static or kinetic friction or charge, but the elasticity was set to .55. This number derives from a ratio that defines the elasticity. The elasticity is found by the minimum bound of the ball (134.6 cm) over the designated height it is dropped from (254 cm) versus the maximum bound (147.32) over the height. We were originally more ambitious, but due to lack of time, the only variable we changed was the radius of the tennis ball. The equation for air resistance, drag coefficient was

with a constant value of .1 kg/m*s. The radii we tested were .06 m, .061, .062, .063, .064, .065, .066, .067, .068, and .07. The proposed range of radius change is actually smaller (between 6.35 cm and 6.36) but Interactive Physics is limited in terms of significant digits, so this was the best alternate.

Interactive Physics offered the best option in terms of testing the theory of enlarging the tennis ball, because we can control the properties surrounding the experiment. There were some problems within the program, however, that caused some error is the entire experiment. For example, although there was an anchored ground, the ball did not always stop directly at 0 m. I.P. let the ball bounce a little bit below or above ground level, because it cannot differentiate small components like that. We took out one of the points (at radius .062m), because it was inconsistent with the rest of the data and the most reasonable explanation is that it stopped at -1.1m, instead of 0m (the rest bounced between -.4 and .6 m, a smaller difference). These were the major obstacles of I.P. that we encountered- limitations in significant digits and measurement- but the data still proved to be informative.

In order to identify the relationship of this data, I tried out various regressions. linear and exponential regressions were surprisingly similar, however. The linear equation was Vf= -187.8r + 48.9, where "Vf" represents the final velocity and "r" the radius. The exponential equation is Vf = 51.22 * .006^r. The linear "r" coefficient is

-.8462, whereas the exponential r coefficient is -.8468. The r coefficient reveals how well data matches with one sort of regression. Partly due to the r coefficient, but also because of the actual graph, we conclude that the exponential regression fits best with the data.

This exponential equation assumes that the graph will asymptote when y=0, however. It would make sense that the data would asymptote at a certain point, but if y asymptotes at 0, then the final velocity would be at 0 m/s. Therefore, for the y to asymptote at a velocity still fast enough for the game of tennis to continue, the radius can only be so big. The data points not only support this conclusion but also reveal the errors in our data. As shown by the graph, some points are further away from the regression lines, while others are closer. This relates to the Interactive Physics problem in differentiating between small numbers. For example, (.063m, 37m/s) first bounced at -1.1m, which is considerably far away from the 0m ground level. (.068m, 36m/s) bounced exactly at ground level and is therefore closer to the line of best fit. It makes sense that as the radius increases, the final velocity decreases, because there is more air resistance attached to the ball. If there had been more time, we would have extended the project by testing other variables, such as air resistance. We also could have left Interactive Physics and tried another method of testing the theory of increasing the radius of the tennis ball in order to slow down the game of tennis. From our extensive research and experimenting, however, we did prove this theory. Hopefully other scientists and researchers dedicated to this project will find similar results and a solution to this rising issue in the tennis world.

Brody, Howard. "The Physics of Tennis." American Association of Physics Teachers, 1979.

Brody, Howard. "Physics of the Tennis Racket II; The Sweet Spot." American Association of Physics Teachers, 1981

Brody, Howard. "The Physics of Tennis III; The Ball/Racket Interaction." American Association of Physics Teachers, 1997

Dyer, Nichole. "Smashing Physics." Science World, September 2001

Hacke, Steve. "Games, Set, and Slower Match." www.physicsweb.com, June 1999

Highfield, Roger. "Larger ball can prevent tennis self-destructing." Daily Telegraph, February 1995

Pallis, Jani. "The Flight of the Tennis Ball" www.Tennisserver.com

The Aerodynamics-in-Sports Team. "Interview with Howard Brody." Cislunar Aerospace, inc. 1997-1999.