Let’s open the book on the physics of the ball during a tennis match once again.
In the first part of this analysis, we focused mostly on the physics of bounces, which is also the only one that can be reproduced in the lab, as well as the easiest to systematise. Therefore, it’s also easier to draw conclusions from it: we explained that the use of heavy topspin reduces the difference between surfaces, even if they have appreciably different physical features in terms of friction and of their coefficient of restitution. However, as many readers have rightly pointed out, it’s not only a question of bounces. The overall behaviour of the ball is influenced by other factors. Tournaments use different balls; the same surface can be made with different deep layers that alter the behaviour of the ball. But above all, weather conditions can change: temperature, pressure, humidity, altitude. That’s the topic we’ll address in this article. We will start with a brief premise on the forces that determine the trajectory of the ball, and then we’ll look at the impact of weather conditions. In the third and last article, which will close this mini-series on physics applied to tennis, we’ll make some considerations on the clay, the surface on which the tours are currently being played.
PART 1: TRAJECTORIES
Although it may seem easy to simply hit the ball in the court, from a physical point of view there is a very limited range of angles and speeds at which the ball can climb over the net and land in the opponent’s court. It must be remembered that the court is less than 24 meters long (23.77): the distance between the net and the baseline is therefore 11.887 meters, while the net is 0.914m high in the middle and 1.07m high at the posts, respectively. Our cerebellum and our visual-motor coordinative abilities do not know how to solve the necessary equations, but they empirically make the calculations needed to successfully hit the ball.
Let’s cap this with an example: for a ball hit from one meter in height from the baseline and at about 108 km/h, the required angle to keep it in play ranges from 4.1° (with respect to the horizontal plane) if hit without any rotation, and of 6.4° with light topspin (about 1200 rpm). When comparing the two ranges, we can gauge one first intuitive truth: a topspin shot is safer, or, in other words, has a greater margin of error. We can add that, for example, serving from the corners rather than from the middle of the court only appears to be more difficult. As a matter of fact, while it’s true that the tract of net to climb over is slightly higher, it is also true that the distance from the service box increases, and, with it, the number of potential hitting angles.
However, the matter is more complex than that. The data of the first example come from an ideal model in which air friction isn’t taken into account. But air friction does exist, it has a great importance, and complicates everything.
A BIT OF PHYSICS (PLEASE DON’T YAWN)
In order to calculate precisely the trajectory of a body in flight, we must examine the horizontal and vertical components of the movement (expressed as horizontal and vertical accelerations). Conceptually speaking, it’s similar to the physics of bounces – however, in that context we only talked about horizontal and vertical speed. The equation of horizontal (Eq.1) and of vertical accelerations (Eq.2) involve several variables, including the radius of the ball (R), its speed (v), the density of the air (ρ) and the coefficient of air resistance (CD, or drag coefficient). The variables combine in the equations (3 and 4 in the following chart) that will be used to calculate the FD and FL, the forces better known as ‘Drag force’ and ‘Lift force’: they are the two components that express air friction both parallel to the body’s motion (FD) and perpendicular to it (FL). It’s quite complicated, but through this diagram we can get a visual idea of the forces involved. Let’s add that, in the equation of vertical acceleration, the gravitational attraction (mg) comes into play, and it’s the only value that depends on the mass of the ball.
To put it simply, a ball hit by a racquet is a body that tries to travel through the air, which in turn offers resistance, since it is a full-fledged fluid with its own weight (a medium-sized room full of air would weigh up to 100kgs). This resistance (expressed by the CD coefficient, that we have already mentioned) is generally independent of the small differences in speed between one shot and another at the usual speed of a tennis match and is therefore more or less constant.
However, things change once again if we consider the rotations and the CL coefficient (lift coefficient), because Magnus effect comes into play. When a ball penetrates through the air, the air itself rushes in and fills the space left empty by the ball. If the ball is spinning, however, the airflow behaves differently. In the case of a topspin shot, the air is deflected upwards, creating a sort of void that pushes the ball downwards, thus making it land a little shorter and keeping it in play more easily. In the case of a backspin shot, the opposite is the case: the air is pushed downwards and therefore the ball receives a push upwards, which leads it to ‘float’ more in the air and thus to land longer. This is one of the reasons why defensive slice shots are more effective (they are more likely to go over the net than topspin shots) but offensive slices are more difficult, because the ball tends to run away lengthwise.
In addition, the uneven surface of tennis balls contributes to the Magnus effect when compared to a smoother sphere such as a baseball. The fuzzy hair of a tennis ball also contributes to normalize the drag coefficient, that we have already taken as more or less constant at the usual speeds of a tennis match. As the final result, it’s a bit easier for a tennis player to guess the trajectory of a shot when compared, for example, to the predictions that a catcher has to make – Jaden Agassi will have to work harder than his dad Andre: luckily for him he is a pitcher, and he has probably inherited a first-rate hand-eye coordination.
