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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Danielle Collins blasts past Iga Świątek and into the Australian Open final

Danielle Collins comprehensively beat Iga Swiatek to reach the Australian Open final.




Danielle Collins (@AustralianOpen - Twitter)

American Danielle Collins made light work of Iga Świątek to move into her first Grand Slam final.


The Pole looked exhausted, particularly in the second set, after her exploits in the quarter finals against Kaia Kanepi, and was no match for the explosive Collins. Świątek going down 6-4, 6-1.

Collins will face Ash Barty in the title match, who also came through in a comfortable straight sets, against another American, Madison Keys, 6-1, 6-3.

It’s looking to be a procession for the world number one in Melbourne, who has yet to drop a set.

On Saturday, the two-time Grand Slam winner will look to become the first Australian women to win on home soil for 44 years.

Having recovered from endometriosis last year, Collins’ run to the final is even more spectacular.

She immediately stamped her authority, breaking Świątek in the opening game.

This was backed up with a comfortable hold, that was sealed with a barnstorming backhand drive. Collins soon nabbed the double break and raced into a 4-0 lead.

But Świątek, to her credit, battled back, holding serve and breaking the American with some explosive hitting.She now trailed 4-2.

A topsy-turvy set of tennis saw Collins break the Pole for the third time, but the drama was only just getting started.

Świątek miraculously saved three set points, the second with a sublime backhand volley, to the delight of Rod Laver Arena, and broke the American, again.

This was backed up with a hold serve, to beg the question, could Collins serve it out and she did, converting her fourth set point in a marathon rally, 6-4.

The 27th seed had all the momentum going into the second set and clinically broke the Pole with some irresistible hitting.

More power tennis flowed from the American’s racquet and she secured the double break, moving into a 4-0 lead.

The seventh seed simply had no answer to Collins’ dominance, and although she saved a match point, the American was far too strong and made no mistake on her second.

After the match she spoke to the crowd.

“It feels amazing. It’s been such a journey and it doesn’t happen overnight. It’s been so many years of hard work and hours at an early age on the court,” she said.

“Yesterday I was talking about all the early mornings where my dad would get up with me and practice before school.

“It’s just incredible to be on this stage, especially with the health challenges, and I’m just so grateful. I couldn’t be happier.”

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Nick Kyrgios Refuses To Engage With Doubles Player’s Criticism After Reaching Doubles Final With Thanasi Kokkinakis

Nick Kyrgios and Thanasi Kokkinakis are into the men’s doubles final at the Australian Open.




Nick Kyrgios and Thanasi Kokkinakis (@AustralianOpen - Twitter)

Nick Kyrgios has refused to engage in criticism from Michael Venus after he reached the Men’s doubles final with Thanasi Kokkinakis.


The controversial Australian reached his first grand slam final with good friend Thanasi Kokkinakis as they defeated third seeds Horacio Zeballos and Marcel Granollers 7-6(4) 6-4.

The Australian duo have also knocked out top seeds Nikola Mektic and Mate Pavic as well as sixth seeds Tim Puetz and Michael Venus.

Speaking of Michael Venus it was the New Zealander who had a problem with Kyrgios’ behaviour in their match branding him as ‘an absolute knob’ as well as stating he has the maturity of a 10 year-old.

After the match Kyrgios refused to hit back at the doubles specialist as he wanted to focus on the victory, “Michael Venus, I’m not going to destroy him in this media conference room right now,” Kyrgios said in his post-match press conference.

“But Zeballos and Granollers are singles players. They’ve had great careers. I respect them a lot more than I respect Michael Venus. I think the balance was there today. The quality of tennis was amazing. I think the festival atmosphere was still there. I think they embraced it. They knew it was an incredible atmosphere.

“Zeballos took a selfie with us before we walked out. That’s how you embrace an atmosphere. You’re not losing a match and then getting salty about it afterwards. It’s ridiculous.”

Kyrgios and Kokkinakis’ reactions have caused a stir among opponents with their over-the-top celebrations after points.

However Kokkinakis told journalists that they are not disrespecting their opponents, “I think for the most part it’s not us trying to disrespect the opponents,” Kokkinakis said.

“It’s us trying to get the crowd going to just increase the atmosphere. Sometimes the opponents take it personally. That’s what happened with the Croatians that we played, the No 1 seeds. That’s obviously Michael took offence to that.

“We’re not doing anything directly to them to try and disrespect. We’re just trying to get the crowd even more hyped, and then some of them take it personally.”

Regardless of their reactions, Kyrgios and Kokkinakis have put a lot of attention on doubles as they bid to win their first grand slam title.

On Saturday they will have an all Australian final with Matthew Ebden and Max Purcell after they knocked out second seeds Rajeev Ram and Joe Salisbury 6-3 7-6(9).

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Ash Barty Storms Into Australian Open Final, Ends 42-Year Wait For Home Country

The world No.1 said it is ‘unreal’ that she now has a shot of winning the title.




Ash Barty has broken new ground at the Australian Open by producing an emphatic win over Madison Keys to reach the final for the first time in her career.


Bidding to become the first Australian women to reach the title match in Melbourne Park since Wendy Turnbull back in 1980, the top seed powered her way to a 6-1, 6-3,win over her below-par American rival. Keys is a former top 10 player who was aiming to reach her first major final since the 2017 US Open. Barty dominated play with the help of 20 winners as she impressively won 78% of her first points. It is the third time in her career she has reached a Grand Slam final.

“It’s unreal. It is just incredible. I love this tournament and I love coming out here and playing in Australia,” Barty said afterwards. “As an Aussie we are exceptionally small but we are a Grand Slam nation and get to play in our backyard.’
“I’m just happy that I get to play my best tennis here. I have done well before and now we have a chance to play for a title – it’s unreal.”

Despite carrying the hopes of her nation on her shoulders, Barty settled into her semi-final clash instantly and displayed no sign of either nerves or tension. She launched her first attack in the opening game by using her slice to apply pressure directly onto Keys’ serve. A drop shot from the American was punished by Barty who hit a cross-court winner to break. She went on to secure a double break for a 4-1 lead by hitting a shot towards the baseline which forced an error from her opponent.

As for Barty’s own serve, she dominated throughout the opener by winning 15 out of 20 points played. It took just 26 minutes for the Australian to secure a 6-1 lead after she returned a tentative Keys serve with a forehand winner down the line.

Inevitably, as the match progressed, Barty looked more tight on the court with the prospect of ending Australia’s long wait for a home player in the final of their biggest tennis event. Nevertheless, she continued to weather the storm before going on to secure a vital break midway through the second frame. After saving a break point in the previous game, a winning Barty passing shot secured another break in her favour to move ahead 4-2. Storming towards the finish line, a serve out wide that was returned out by Keys set her up with two match points. She prevailed on the first of those with yet another forehand winner.

“The conditions were different tonight. It was humid and the ball was a little bit heavier off the strings. I just tried to run and adapt, make as many balls as I could and keep Maddie (Keys) under the pump on her serve,” Barty reflected.
“It was important to stay point-by-point and do the right things each and every time.”

There is a silver lining for Keys who exits Melbourne with a surge in confidence after what was a troublesome 2021 season marred by injury and a loss of form. Last year she only managed to win 11 matches on the Tour compared to this month where she has been able to win 10 alone. Keys has been ranked as high as seventh in the world.

“It’s nice to see her back where she belongs. She is an amazing human being. You see the way she carries herself out on the court,” the two-time Grand Slam champion said of Keys.
“The thing I love most about Maddie is that she is a great person no matter what happens on the court.”

Barty run to the final is yet another success story for Australian tennis this year. Nick Kyrgios and Thanassi Kokkinakis are through to the men’s doubles final where they will play compatriots Matthew Ebden and Max Purcerll. Australia also has representation in the mixed doubles final with Jamiee Fourlis and Jason Kubler. However, Barty says the highlight of her country’s success is that of wheelchair tennis star Dylan Alcott, who played his last match today before retiring.

“Dylan for me is in front of that. He’s inspired a nation and the whole globe,” she said.
“The way he and the Australian Open have worked together to open up the opportunities for more disabled people around the world to play tennis is exceptional and I couldn’t be more proud of him.”

Barty will take on either Danielle Collins or Iga Swiatek in the title match. She leads Collins 3-1 in their head-to-head and has won both of her previous Tour meetings against Swiatek.

The last Australian woman to win the Australian Open was Chris O’Neil in 1978.

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