Surfaces are not all the same. Every now and then someone surmises that this is not the case anymore, but there are data to show that differences that once seemed more evident still exist today.
Grasscourts have become slower than a couple of decades ago, this is confirmed, but the distance between clay and hard doesn’t appear to have altered. After all, while hyper-champions like Djokovic and Nadal have found a way to excel even on the surface that offers them fewer natural advantages, many others continue to struggle on courts that do not suit them. It turns out that hardcourt tennis is not identical to claycourt tennis, because the physics of shots are just different. If that weren’t the case, the performance of all players would be uniform, matches would be more or less the same, and the aces data – the best metric to indirectly compare the speed of the courts – would remain in the same range on all surfaces. Instead, Nadal’s career data state that the Spaniard hits an average of 2 aces per game on clay and 3.5 on hard courts (+75%); Federer 5.9 and 7.9 respectively (+34%); Djokovic 3.7 and 5.6 (+51%).
It follows that on clay, the elective surface of this part of the season, tennis changes. But how exactly? What is the best way to move on claycourts, and what is the yield of the shots as compared to other turfs? What are the difficulties that players face compared to hardcourts, and what are the advantages? It’s a long and complex issue, but before going deep into tactical considerations – where a bit of subjectivity takes over – we will explore the problem from a physical and theoretical premise in order to build a solid starting point for our digression. We will draw our conclusions in a second article, which will be published in the coming weeks.
Special thanks for this article must be paid to Matthew Willis, who curates a very interesting tennis blog on Substack. He got us to discover the publications of Rod Cross, a former Physics Professor at the University of Sydney who has dedicated a large part of his career to the subject of physics applied to sports. If the topic appeals to you, some of his work can be found here, here and here. In the next sections we will try to summarise the main discoveries of his research, broken down into simple concepts. If you are not interested in physics and prefer to go directly to the conclusions, you might want to go straight to the second section.
FIRST SECTION: THE THEORY OF SURFACE BOUNCE PHYSICS
To put it very simply, the bounce of a tennis ball can be viewed as a physical system in which a spherical body has a speed that can be divided into two components: horizontal inbound velocity (vx1, it stems from hitting the ball with the racquet, measured in m/s) and the vertical inbound velocity (vy1, it stems from the stroke and ‘fights’ with gravity, measured in m/s). After the impact with the surface at a certain angle of incidence (θ1), the two outbound velocities are obviously reduced (vx2 and vy2); this means that the ball loses some of its thrust and speed and loses a little more or a little less depending on the surface on which it bounces.
Graphical representation of ball bounce (credit to Rod Cross)
Let’s start from the beginning. In 1984, Howard Brody developed the first model to study the physics of the tennis ball, imagining it as a rigid object which after the impact with the surface does not deform. This model, which turned out to be inaccurate and incomplete, assumes that the horizontal outbound velocity of the ball after the rebound is always 64.5% of that before the rebound, regardless of the surface and of the angle of impact, as long as it is greater than 16 degrees.
In reality, the ball deforms after the impact. This is why the physics of the bounce are much more complex (“the worst calculus final you ever had in your life,” according to the late, great David Foster Wallace) and consequently not all surfaces are the same. For a few fractions of a second, in fact, the ball – which possesses with a certain rotation – stops rotating and begins to slide on the court, covering a micro-distance (D as showed in the above figure) which corresponds to the displacement of the axis of force N, i.e. the one that fights with friction (F) to push the ball upwards. After this transitional phase, the ball resumes the rotation motion and takes off towards the phase following the rebound. The duration of this phase, and therefore the resistance that the surface offers to the ball, depends on the friction of the surface itself and the type of shot (we will get to that shortly). On the clay it lasts a little longer, so the distance D is greater and the surface “steals” more inertia from the slowed ball that comes out; on hardcourts it lasts less, so the distance D is smaller, and the ball resumes its upward motion more quickly, resulting faster after the rebound. Thus, the 64.5% rule fails.
The premise is completed by specifying that two physical characteristics are attributable to the surface:
- The coefficient of friction (μ), which measures the friction of the surface by subtracting the horizontal outbound velocity (vx2 after the bounce) from horizontal inbound velocity (vx1). Basically, it tells us how much speed the ball loses in the horizontal plane. The higher this coefficient is, the more the surface generates friction (as happens on clay) and consequently slows down the stroke.
- The coefficient of restitution (e), which instead measures how much the surface “helps” the ball to bounce and is the ratio between vertical outbound velocity (vy2) and the vertical inbound velocity (vy1). The higher this coefficient is, the more generous the surface is with the bounce (like on clay).
If it is easier to understand why the increase in the friction coefficient slows down shots (and therefore the game), it may be necessary to specify why a court that ‘returns’ more is considered slower: a higher bounce gives the player more time to execute the stroke and to find the ideal sweet spot for impact, whereas low bounces force the receiver to react in a shorter time span.
Lorenzo Musetti (Acapulco 2021 / photo AMT): example of impact below the level of the hips
These two physical characteristics have been incorporated into a formula developed by the ITF to calculate the Court Pace Rating (CPR), an indicator of the speed of the courts. The formula is the following:
This rating, which basically tells us how much speed the ball has before bouncing and how much speed it has afterwards, is calculated in a laboratory under fixed conditions. Basically, a shot is hit at about 67,1 mph on a sample of the surface, without topspin and at a fixed angle of 16 degrees. However, this is a partial figure, because it does not take into account what happens when the ball hits the surface at a greater angle, namely when the stroke is executed with topspin – as previously implied, when the ball comes in with a lot of rotation, things are different. The above metrics is also not totally representative because it doesn’t take into account other factors that influence the speed of the court, such as weather conditions and all the layers composing the court beneath the upper surface.
CPR should not be confused with CPI (Court Pace Index), which is based on the same physical premises but is not calculated under laboratory conditions – it is simply inferred from the speed measures offered by Hawkeye data (used at Slam, Masters 1000 and ATP Finals levels). In some ways, it is a more truthful measurement, as it is based on matches data from played tournaments and on a larger sample of shots.
