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Under a (probably) cloudy sky the eyes of a silent, expectant crowd – sated with Pimms and strawberries and cream – fixed on the ball as it was gently tossed from the server’s hands and arced in a gentle parabola; the whisper of a collective intake of breath broken by a sharp ‘toc’ as the tennis ball made contact with the onrushing racket and went hurtling towards the opponent. People have been watching a similar sight since real tennis came into being in the Middle Ages, derived from the French racket-less game of jeu de paume (which also gave rise to handball). One such person was Sir Isaac Newton, who in a paper written in 1672, wondered why it was that a tennis ball was able to follow ‘such a curveline’ as it went from its origin to its destination.



The game of tennis as Wimbledon crowds know it today was invented by Major Walter Wingfield in 1873 and a few years later Lord Rayleigh followed up on Newton’s musings in his paper On the Irregular Flight of a Tennis Ball, in which he sought an explanation for why ‘a rapidly rotating ball moving through the air will often deviate considerably from the vertical plane’. In it, he credited the work of the German scientist Gustav Magnus, who in the mid-19th Century was studying the deflection of projectiles from firearms.

A ball with spin will rotate, with half of the ball rotating in a direction opposed to the incoming air flow and the other half rotating in the same direction as the flow. Bernouilli’s Equation relates pressure and velocity according to the formula $$p +\frac{1}{2}\rho v^2 = \mathrm{constant\ (along\ a\ streamline)},$$ where $p$ is pressure, $\rho$ is fluid density and $v$ is velocity. The side of the ball that is spinning into the incoming air will cause the air to slow down and Bernouilli’s Equation tells us that this will result in an increase in pressure ($v$ decreases, thus $p$ must increase). The opposite occurs for the side of the ball spinning in the same direction as the air: the air speeds up and pressure decreases. Hence we have regions of higher and lower pressure, causing the ball to veer in the direction of least pressure. Such a phenomenon is known as the Magnus Effect and provides us with a simplified description of the ball’s motion.

In 1904, the aerodynamicist Ludwig Prandtl discovered that a thin layer of fluid, known as a boundary layer, exists on the surface of a moving object: it is the separation of this layer from the tennis ball (occurring earlier if the ball is moving into the fluid, later if it is retreating, as in the picture below) that results in pressure differences and gives rise to the Magnus Effect.

Despite being able to serve at phenomenal speeds (the fastest ever serve at Wimbledon was the 148 mph achieved by Taylor Phillip Dent in 2010), tennis players have to make use of the Magnus Effect to add unpredictability to their serves and keep their opponents on their toes. A stroke on the side of the ball with the racket when serving will make the ball rotate around a vertical axis, known as a slice serve, and cause it to follow a curved path as it flies through the air. Adding top spin on the other hand by arching the racket to brush up the back and over the top of the ball will result in the ball rotating about a horizontal axis and dropping more quickly, giving it a higher bounce when it lands on the other side. Tennis players will in general make use of both top spin and slice at the same time when serving, trying to bamboozle their opponents with the flight of the ball and its bounce.

Faster than the eye could see, the tennis ball flew through the air. Lost in a blur of green, it curved majestically around, reaching its maximum perpendicular displacement from the vertical plane between server and intended target as it zipped over the net. Before the opponent had time to move, before the heads of those in the crowd had, en masse, snapped round from left to right in a vain attempt to keep up with the ball, the ball completed its precipitous drop and thumped against the pristine grass, spinning away in mockery.

Dr Nick Ovenden Dr Nick Ovenden is a senior lecturer in Mathematics at University College London, currently focusing on developing mathematical models of problems from medicine and industry.

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