Mathematics of racquet sports
The Mathematics of racquet sports encompasses the application of mathematical concepts to analyze equipment, gameplay, and scoring systems in various racquet sports, including tennis, badminton, squash, and table tennis. Mathematical modeling is utilized to understand factors such as ball spin, trajectories, and the dynamics of racquet designs, where aspects like weight distribution and string material significantly affect performance. The scoring systems in these sports are often non-linear and can lead to scenarios where a player may win more individual points yet still lose the match, highlighting the complexity of competitive play.
Moreover, computer-aided design plays a crucial role in optimizing racquet performance, allowing for rapid assessments and adjustments. Training for athletes also benefits from mathematical analysis, particularly in studying muscle strain and developing nutritional guidelines. In terms of projectile behavior, the performance characteristics of balls and shuttlecocks are strictly regulated, with specific standards set for dimensions and rebound rates. As such, mathematics serves as a foundational tool in enhancing both the strategic and physical aspects of racquet sports, enriching player performance and game analysis.
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Mathematics of racquet sports
SUMMARY: The equipment, game play, and scoring of racquet sports can be analyzed using mathematical concepts, such as vector operations and probability
Racquet games include sports such as tennis, badminton, squash, and table tennis, as well as other less popular games like real tennis, racquets, and racquetball. Mathematics has many roles to play in these games. These range from equipment testing and court marking to training and analysis of play.
For example, the scoring system in tennis is not a simple counting or linear progression. Mathematicians model a ball’s spin in multiple axes, along with trajectories and deflections, as functions of other variables. Markov chains and vector operations can be used to analyze the progression of games and both probability and statistical methods are used to describe performance, seed players for competition, and predict outcomes of matches.
Racquets
Racquet weight distribution, shape, and string material are important factors in the resultant power, accuracy, and comfort of a racquet. Increasing power, for example, can lead to a decrease in accuracy and it is important to balance these properties. Computer-aided design is the natural choice for this process because of its fast and powerful recalculation abilities.
Projectiles
Racquet sport projectiles such as balls and shuttlecocks are subject to strict regulations and must adhere to these for as long as possible at the highest levels of play. For example, the World Squash Federation allows balls that are 40 millimeters in diameter and each must be tested at 23 degrees Celsius (73 degrees Fahrenheit) and 45 degrees Celsius (113 degrees Fahrenheit), room temperature and play temperature, respectively. There are several dot grades according to level of rebound but an average squash ball rebounds at around 30 percent (dropped from a height of 3.2 feet, it should reach 12 inches on the bounce). A tennis ball rebounds at around 50 percent, although changes in ambient air pressurebecause of altitudecan affect this figure. Table tennis balls rebound at 85 percent.
A popular way to gauge the overall performance of these projectiles is to measure their maximum speed. Tennis balls appear to hold the record for being the fastest, and indeed, the now-retired, Andy Roddick could propel a tennis ball up to 155 miles per hour. As recently as the 1980s, professional tennis players still used wooden racquets. Their subsequent replacement by those made of graphite fundamentally changed the game. Using the new graphite racquet, players also learned to flick their wrists so that the face of their tennis racquets did not impact squarely with the ball. This reduced the friction between the ball and the racket and generated more torque, or spin, on the tennis ball. This spin allowed the ball to travel in a straighter line and a return serve by a player could be more accurately aimed. This is a similar principle to how a bullet leaves a rifle. A groove inside the barrel of the weapon, or its rifling, allows the bullet to develop a spin, keeps it on a tighter trajectory, and hence increases its accuracy. Players also use spin to change the ball's trajectory and to make their returns more difficult for an opponent to counter. By contacting the ball on its top half, topspin can be generated. The ball rotates in a forward motion and the trajectory will first arc and then drop. Following these types of advancements, power and spin became more emphasized in tennis, and the sport developed beyond a game of finesse.
Despite the speed now generated by tennis players, badminton can be even faster. A badminton stroke can attain a velocity of over 186 miles per hour. This figure seems counterintuitive because a shuttlecock slows down much more quickly than a tennis ball.
Training
One of the most important roles for mathematics in racquet sports is in training. Sports science researchers study muscle and joint strain and develop nutritional guidelines that allow the player to remain comfortable and energetic during play. Of the racquet sports, squash is regarded as the most intense as players burn roughly 50 percent more calories per hour than badminton or tennis. However, tennis games can run several hours, whereas badminton and squash games are typically decided in under an hour. The total number of calories burned is the product of the calories per hour and the number of hours.
Scoring
In all of the major racquet sports, and many others, a feature of the scoring system may mean that the player who wins more individual points or rallies can still lose the match. Consider the scores of the 1972 British Open final decided by the best of five games, each played to nine points: 0–9, 9–7, 10–8, 6–9, and 9–7. The loser, Geoff Hunt, scored 40 points and won two games. The winner, Jonah Barrington, scored 34 points, won three games and the title.
The same quirk appears in any scoring system where victory is decided by the most wins over a specific number of games. In tennis, this feature exists on two levels. It is possible to win more points and more games but still lose the match. For example, if a match ends 6–4, 0–6, 6–4, 0–6, 6–4, the winner wins 18 games, the loser wins 24 games. The maximum difference in points or rallies in this case is 60 (72–132) in favor of the loser.
Bibliography
Gallian, Joseph. Mathematics and Sports. Mathematical Association of America, 2010.
Havil, Julian. Nonplussed! Mathematical Proof of Implausible Ideas. Princeton University Press, 2007.
Jackson, George. "What Forces Are Involved in Tennis?" Physics Network, 15 May 2023, physics-network.org/what-forces-are-involved-in-tennis. Accessed 17 Oct. 2024.
Kellog, Colton. "Serving Up Some Knowledge: The Physics of Tennis." University of Southern California, 9 Apr. 2019, illumin.usc.edu/serving-up-some-knowledge-the-physics-of-tennis. Accessed 17 Oct. 2024.
Lees, A., D. Cabello, and G. Torres, eds. Science and Racket Sports IV. Routledge, 2009.
Lees, A., et al., eds. Science and Racket Sports III. Routledge, 2004.
Sadovskii, L. E., and A. L. Sadovskii. Mathematics and Sports. American Mathematical Society, 2003.
"What Is Topspin And How Is It Used In Tennis?" Tennis Central, 2022, tennis-tennis.com/what-is-topspin-and-how-is-it-used-in-tennis/. Accessed 17 Oct. 2024.