Tennis And Physics: On Trajectories And The Heat Factor - UBITENNIS
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Tennis And Physics: On Trajectories And The Heat Factor

Contrary to public belief, humidity has very little effect on ball speed. What really matters are the heat and especially the altitude at which a match takes place



Novak Djokovic - US Open 2018 (credit USA Sports)
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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.


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.


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.

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Simona Halep Provides Injury Update On Knee Ahead Of US Open

The return of Simona Halep looks a long way away from happening as a knee injury continues to bother the Romanian.



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Simona Halep has provided an update on her knee injury as she faces a race against time to be fit for the US Open.

The former Roland Garros and Wimbledon champion’s comeback has not gone according to plan so far as she returned from a doping ban in Miami earlier this year.

However playing time has been limited for the Romanian as her second tournament of the year ended in a first round retirement at a Challenger event in Paris.

As a result of her retirement Halep was forced to miss Roland Garros and Wimbledon as she looked to recover from a knee injury.

The injury is more serious than meets the eye though as Halep provided an update on her knee injury and revealed it’s an injury that has been occurring for the last few months, “I wanted to provide a little health update since I have been away from the court lately. It is not the best news, but I wanted to share all the same,” Halep revealed on Instagram.

“For a few months now, I have been struggling with a knee injury. It’s an injury I have never had before and it has been a tough one to manage, with the pain persisting. It has been a very difficult period (almost two years) mentally and emotionally for me and struggling physically on top of that definitely doesn’t help.

“I have decided to take the time needed to recover properly, rather than training and playing through pain. My instinct has always been to try to come back as soon as possible, but I am not a machine, I am a human being and I need time to recover from everything I have gone through.

“I’ll keep you posted and, as always, want to thank you for your permanent support. I hope to be back on court very soon.”

Halep now faces a race against time to be fit for the US Open which starts on the 26th of August.

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Alexander Zverev Commits To Olympics, Speaks About Miscommunication Between Olympics And ATP

Alexander Zverev is preparing for his Olympic title defence.



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Alexander Zverev has spoken out on the miscommunication between the Olympics and the ATP tour with tennis’ top stars set to miss some big events on the tour.

The German is getting ready to defend his Olympic title that he won in Tokyo as Zverev heads to Paris.

After narrowly missing out on Roland Garros glory, Zverev now has an opportunity at redemption as he aims for a second consecutive gold medal at the Olympics.

However there is a dilemma for big players this year as the same week as the tennis event at the Olympics at the ATP 500 event in Washington.

This is an event that Zverev usually plays but will have to miss it due to playing in Paris.

Speaking out on the miscommunication, Zverev said it’s no secret that the ATP are not best friends on the Olympics but has committed to the historic event, “The ITF or the Olympics are not best friends with the ATP, that’s not a secret,” Zverev was quoted by Tennis Infinity as saying.

Everybody continues their own tour. Yes, the Olympics is one of the, if not the, most important events we have in sports. So, the best players in the world are always going to play the Olympics over anything else, over points and over prize money.

All the top players went to Wimbledon a few years back, despite having no points on the line. It’s about prestige, it’s not about points or money. It’s about winning these kinds of titles and having medals for your country.

That’s what it’s about. I would never choose to play a 500 event over the Olympics. There are no thoughts on that, and I think Washington is going to be quite an interesting 500 event, I would say.

The ATP 500 event in Washington will be headlined by Grigor Dimitrov, Ben Shelton, Karen Khachanov and Frances Tiafoe as they are among those that have chosen ranking points over the Olympic experience.

As for Zverev he is continuing his Olympic preparations in Hamburg where he will face Pedro Martinez in the semi-finals.

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Rafael Nadal Reveals Delight At Setting Up Ajdukovic Semi-Final After Four Hour Victory

Rafael Nadal has revealed his delight at reaching the Bastad semi-finals after a four hour victory over Mariano Navone.



(ubitennis/Francesca Micheli)

Rafael Nadal has revealed his delight at reaching the semi-finals in Bastad after a four hour victory over Mariano Navone.

The 22-time Grand Slam champion prevailed in a marathon 6-7(2) 7-5 7-5 victory to reach his first semi-final since Wimbledon 2022.

The Spaniard converted ten of his 21 break point opportunities as he reached the semi-finals.

After the match Nadal revealed his delight at reaching the last four and prevailed Navone for his resilient display, “I lost for some moments my concentration, but I was able to hold physically until the end,” Nadal told the ATP website.

“That is so important for me. Let’s see how I am tomorrow, but today I am alive and in the semi-finals, so that’s very important. There were a lot of changing dynamics in every single set. For moments, he was in control. For moments, I was in control. But at the end, no one was in control! That’s true, and I had a good chance in the second with 3-0. Then I was very close to losing the match in the second set.

“In the third [I was ahead] again with 5-2, but he’s a great fighter and he played a great match. I wish him all the very best for the rest of the season.”

Nadal will now play Duje Ajdukovic in the semi-finals as he aims to continue his eight match winning streak in Bastad with this being his first appearance at the tournament since winning the title in 2005.

The other semi-final will be Nuno Borges against Thiago Agustin Tirante with both players reaching their maiden ATP tour semi-final.

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