The Physics and Perception of Court Speed

What’s big, flat, never moves, and yet has a speed debated hotly by thousands of people? Tennis fans know the answer to the riddle is a court surface.

Fans, players, coaches, and pundits often have strong opinions about court speed. As powerful servers increasingly dominated the game in the 1990s, many insisted the faster surfaces needed to be slowed down to preserve rallies and level the playing field. Today it’s fashionable to bemoan the “homogenization” of tennis courts, and there’s a growing belief that every tournament’s surface plays like “fast clay” or “slow hard court”, reducing the variety in playing styles. Some such criticism is based on the hasty impressions of fans who think they know the speed of a court after watching one set on a fuzzy internet video stream—even when that impression is as physically implausible as that of an aspiring journalist who once told me that the indoor hard court at the WTA Championships in Istanbul played slower than the red clay at Roland Garros.

At the same time, players who play and practice on these courts day after day continue to perceive significant differences in surfaces that are theoretically identical. Last year, Paris-Bercy tournament director Guy Forget insisted the court surface at his event was deliberately made to be exactly the same as that used the following week in London for the ATP World Tour Finals. Rafael Nadal said the court in London was slower. Novak Djokovic said it was faster. How is this possible? Are the players imagining things? Are the tournament directors lying? Or are there other factors besides the courts themselves that come into play?

Can Court Speed Be Measured?

Like many terms in tennis, court speed—or more broadly, the speed of playing conditions—is a phrase that’s widely used but whose precise meaning is often left unclear. Is a fast court one that favors shorter points, leading matches to be completed in less time? Is it one that favors playing styles (serve and volley!) traditionally associated with fast courts? Is it one on which players serve more aces? Is it one on which balls retain more of their initial speed after they bounce? Is it one on which balls bounce lower? Generally speaking, all of these factors are related, but they are not the same, and in some circumstances they can even work against each other.

Some readers will be aware that the International Tennis Federation measures and classifies the speed of various surfaces. These ratings are most often seen in Davis Cup and Fed Cup, where they give visiting players and fans a rough idea what to expect from the court the host team has selected. The ITF calculates these ratings using this formula:

Court Pace Rating = 100(1 – µ) + a(be)

Courts that yield a rating of 45 or higher are classified as “fast”, those with a rating of less than 30 are “slow”, and there are three more categories in between.

That seems pretty objective so far, and sounds even better if you know that µ (the coefficient of friction) and e (the coefficient of restitution) are calculated from actual tests where “high-specification balls”, with essentially no spin, are fired at the court surface in a repeatable way at a specified angle using a pneumatic cannon, and the balls’ speed, angle, and spin after they bounce are all measured precisely.

Things get a little sketchier when one learns that b, which is equal to 0.81, is supposedly the average coefficient of restitution “for all surface types”. Which surface types? All the ones the ITF has measured? All the ones that are used in professional tournaments? Is the average weighted to take into account that some surface types are more commonly used than others?

Never mind, because the game is really up when one discovers that a is called the “pace perception constant”, and is set to the very arbitrary-looking value of 150. So after all of this careful procedure with specially selected balls, calibrated pneumatic cannons, laser velocity sensors, high-speed cameras, and probably white lab coats, the final result comes down to a quantified admission that the whole enterprise is subjective?

But, wait. Court speed isn’t merely a matter of perception. Points tend to be longer if defending players have more time to get to balls and do enough with them to prolong the rally. Let’s look at the physics of how different surfaces contribute or take away critical hundredths of a second.

Physics to the Rescue

In the ideal world of many academic physics problems, where objects are rigid and have perfect geometric shapes and collisions are perfectly elastic, balls would retain all their speed when they bounced and leave the court at the same angle they landed. In the real world, tennis balls stay in contact with the court for a significant time as they bounce, and some of their energy is dissipated as they skid and flex and, in some cases, as the court surface deforms. These effects occur on a molecular level in non-homogeneous materials, and are impractical to model precisely. But by indirectly measuring the total forces acting on the ball through the entire duration of the bounce, we can create a simplified mathematical model that approximates the ball’s behavior well enough to resolve most of the mystery about court surfaces. Here’s a diagram of the forces involved:

The two small arrows shown in the center of the ball represent the forces of gravity and aerodynamic drag, both of which are critical to understanding the ball’s trajectory in the air, but which are small enough to be ignored during the bounce, which lasts only about 6 thousandths of a second. The way the ball bounces is determined overwhelmingly by two forces acting where the ball contacts the surface—the force of sliding friction (ƒ), which slows the ball’s horizontal speed and also tends to add topspin (ω), and the so-called “normal” force N, applied by the court and by the ball as it resists compression, stopping the ball’s descent and redirecting it upward. The friction force is equal to N multiplied by µ, the coefficient of sliding friction. N varies throughout the bounce in a way that’s hard to model, but its average value can be calculated from e, the coefficient of restitution, which is obtained simply by dividing the ball’s vertical speed before the bounce by its vertical speed after the bounce. These two coefficients, µ and e, are the key variables in the ITF’s Court Pace Rating formula.

Unfortunately, the ITF simplifies the measurements and calculations it needs to make by basing them on an oversimplified model of the physics involved, which limits the model’s usefulness in understanding how the ball behaves in a typical tennis match. One problem is that it assumes the court surface is perfectly rigid. This is a pretty good assumption with regard to common hard courts, but it breaks down on clay, where the ball digs visible craters as it bounces. This absorbs some of the ball’s energy, but also reconfigures the geometry of the bounce. As the ball rebounds, it must climb out of a depression, which causes it to bounce higher and slower as if it were bouncing off of an inclined surface (see illustration below). Of course, clay courts differ from each other, and the phenomenon is very difficult to model. To some degree, it can be addressed by incorporating its effect into the measurements of µ and e.

The ITF also assumes that the ball stays rigid and spherical throughout its bounce, rather than compressing as I show in the diagram. Even worse, it assumes that the ball slides against the court surface throughout the bounce. In reality, as the friction force adds spin to the ball, the rearward rotation of the bottom of the ball tends to approach the value of the ball’s horizontal linear speed. If these two values become equal, the ball stops sliding and starts rolling like a wheel, and the friction force drops to essentially zero. The more topspin the ball starts with, and the higher the angle at which it lands, the sooner this transition occurs, and the less effect friction has on the bounce. In fact, a shot that lands with enough topspin to roll during the bounce on a slippery surface will rebound with essentially the same horizontal speed as if it had bounced off a higher-friction (slower) surface. This effect turns out to be very important to the way balls bounce in real matches and the way different players perceive court speed. Indeed, the ITF only gets away with ignoring it in their tests because they require the tests to be conducted under conditions where it doesn’t happen—balls bouncing at high speeds (67 mph or 108 km/h, much faster than groundstrokes generally land), low angles (16°), and no initial spin.

I addressed these two most serious limitations of the ITF’s model (using mathematics I’ll spare you from here) and combined this ball-bounce model with an aerodynamic model of the ball’s behavior in flight (which I discuss here) to develop a computer simulation of a tennis shot’s entire trajectory. This allows me to investigate what kind of difference varying the court surface makes to the behavior of otherwise identical shots.

Simulations

To get a feel for the general effects of different surfaces, let’s compare the trajectories of a hard crosscourt groundstroke, after it bounces off of a fast grass court or a slow clay court. First, a “flat” shot hit with no spin: (The vertical scale is stretched to magnify the differences in shape.)

On clay, such a shot loses more of its speed during the bounce than it does on grass, and therefore takes significantly longer to reach the baseline. It also bounces higher, which gives a defending player even more time to reach it before it bounces again or leaves the playing area. Up to a point, the extra height also places it closer to the defending player’s ideal hitting zone.

Now, to show that spin is at least as important as the surface in determining how balls bounce in today’s game, here’s a shot launched at the same speed and landing in the same spot near midcourt, but with heavy topspin:

Compared to the equivalent flat shot, the topspin shot bounces higher. Contrary to a widespread misconception, this is due less to any vertical “kick” the spin creates against the court than it is to the fact that topspin makes the ball curve downward as it flies through the air, increasing the angle at which it strikes the surface. On slower surfaces, the topspin shot also retains more of its speed after the bounce, because it transitions from sliding to rolling motion sooner and experiences less friction than the equivalent shot hit with no spin. Indeed, a topspin shot like this one has essentially the same horizontal speed after the bounce, and very nearly the same speed at the baseline, no matter what the court surface is. On slower surfaces, the topspin shot does stay in the air a bit longer before it goes out of play, but this is due more to the fact that the higher-bouncing shot takes a longer path than due to any difference in speed.

Already we can see that players who use a lot of topspin are unlikely to notice differences in the coefficient of friction, µ between two surfaces (unless a court is so slippery that they fall down while running for the ball). Instead, their impressions of court speed will be dominated by differences in the coefficient of restitution, e. Higher values of e will make their shots bounce higher and stay in play longer, an effect magnified by the fact that their high-topspin shots inherently tend to bounce higher anyway. Players who use less spin will see µ as much more important, because it makes more difference in the speed their shots carry after they bounce.

Can Modern Surfaces Really Be Distinguished From One Another?

The real teeth-gnashing about court surfaces doesn’t start until we discuss the variation in playing speed among surfaces of the same general class, and the extent to which one kind of surface can play like another. To get an idea of how much of such variation and overlap there is, I hunted down measurements of court surface characteristics from several different sources, and diagrammed them below:

As one might expect from a living surface that’s especially sensitive to changing moisture levels and atmospheric conditions, grass shows the most variability, at least in terms of restitution, or how high balls bounce.

“Natural” clay, usually made of either crushed brick (red clay) or crushed stone (green clay), shows relatively little variation, though of course clay is sensitive to moisture levels as well. Green clay is generally thought to play faster than typical red clay, but I can’t find any hard data to confirm this. Indeed, the data indicate that all natural clay courts have playing speeds well within the ITF “slow” category. “Artificial” clay, which is composed partly or entirely of more modern synthetic materials, can play somewhat faster (not shown in the diagram).

Grass and hard courts show the most overlap in terms of their playing characteristics. The ITF even reports that the average court among all grass courts it has tested has playing characteristics within the range covered by hard courts, and the average hard court has characteristics well inside the range measured for grass courts. As I hinted earlier, I’m skeptical of this averaging of results from multiple courts, particularly since we don’t know what fraction of the courts they’ve measured are used in tour-level tournaments. I’m particularly skeptical of their data on grass courts, which is largely inconsistent with measurements from other sources. Their testing methodology may not provide a realistic idea of the way grass courts play as balls bounce with wider ranges of speed and angle in real matches. The ITF has recently classified the playing speed of Wimbledon’s courts as “medium”, which implies they play significantly slower than either the courts at the US Open (fast) or the Australian Open (medium-fast). This is simply not consistent with match results at these events.

There is a narrow sliver of the graph where all three surface types overlap, suggesting it is possible to prepare courts of all three types whose bounciness and friction characteristics are essentially the same. But it seems most clay courts are still significantly slower than the other court types.

Not All “Medium”s Are Created Equal

Perhaps what surprised me the most about these court measurements is how much variation there is in the coefficients of friction between different hard courts. Hard courts span the whole range of the ITF’s five court pace categories, though none of them play as slow as typical clay courts or as fast as faster grass courts. More significantly, hard courts that are all in the same pace category can have very different combinations of characteristics—some slippery but bouncy, some grippy but with a low bounce, and some in the middle.

How much effect do such differences have? Let’s do some more simulations. First, here are the court parameters I used for the simulations below:

Surface e µ Court Pace Rating
Fast grass 0.70 0.55 61.5
Hard (slippery, high bounce) 0.87 0.55 36.5
Typical medium-pace hard court 0.80 0.65 36.5
Hard (grippy, low bounce) 0.73 0.76 36.5
Slow clay 0.88 0.76 13.5

Notice that all three hard courts have identical ITF pace ratings, even though one’s coefficient of friction is the same as that of a slow clay court and another’s is the same as that of a fast grass court. Now, here’s how the same groundstrokes whose trajectories I plotted above play on these five surfaces:

Crosscourt Groundstroke, No Spin
Surface Speed at baseline, mph Height at baseline, ft Time to baseline, s Final speed, mph Total time in play, s
Fast grass 37.2 1.3 0.975 34.7 1.186
Slippery, bouncy hard 36.3 2.1 0.983 32.8 1.310
Typical medium hard 34.7 1.8 0.996 32.0 1.272
Grippy, mushy hard 33.2 1.5 1.010 31.2 1.228
Slow clay 32.1 2.3 1.022 29.5 1.352
Crosscourt Groundstroke, Heavy Topspin
Surface Speed at baseline, mph Height at baseline, ft Time to baseline, s Final speed, mph Total time in play, s
Fast grass 36.6 2.8 0.998 32.0 1.432
Slippery, bouncy hard 36.8 3.9 0.998 31.6 1.448
Typical medium hard 36.6 3.5 1.000 31.6 1.452
Grippy, mushy hard 36.3 3.0 1.001 31.6 1.457
Slow clay 36.5 4.0 1.001 31.3 1.455

Once again the heavy topspin shots, landing at an angle of 18.1°, are rolling rather than sliding by the time they rebound, no matter what the court surface is, and therefore they all carry very similar speeds to the baseline. Even the variations in the total time a defender has to hit them before they go out of play are fairly small. The main difference in how such shots play on the various surfaces are in the height of the bounce—a higher bounce keeps the ball in a convenient hitting zone for the defender for a longer period of time, and therefore contributes to longer, more neutral rallies. Thus in matches involving heavy topspin, players are likely to perceive the slippery, high-bouncing hard court as playing significantly slower than the other hard courts, even though it has the same ITF pace rating as the other hard courts and the same coefficient of friction as the grass court.

The flat shots, landing in the same place in the court at a lower angle of 11.9°, continue sliding throughout the bounce on every surface. This means they’re slowed by friction over a more extended time span, and they show bigger variations in speed. Since they bounce lower, the variations in their height at the baseline are also smaller. Defenders in flat-hitting matches are more likely to perceive the differences between the courts in terms of how much time they have to reach the ball than in how high it is when they get there. At least when playing near the baseline, they are more likely to perceive the grippy, low-bouncing hard court as playing slower than the other hard courts, even though it has the same pace rating as the other hard courts and a coefficient of restitution similar to the grass court.

Blame the String Manufacturers

The influence of topspin on how balls bounce and the extent to which court surfaces affect play during a match is usually overlooked by fans and pundits. Modern synthetic racket strings are more elastic and create less friction where they rub against each other, allowing them to return more of the energy of a player’s swing to the ball, and making it easier for more players to generate more spin. Topspin is defined as spin in which the top surface of the ball is rotating in the same direction as the ball’s linear motion. Therefore defending players have to swing their rackets faster to effectively return heavy topspin shots, because they must reverse the spin their opponents put on the ball in order to flatten out a shot or create topspin of their own. As players start using more topspin, it creates an arms race in which more players adopt slick new strings and make more effort to develop more topspin. The result is that most tour-level matches today occur between players using significant topspin at least part of the time, and in such matches the differences in friction between various surfaces may have little enough effect to go unnoticed by players and fans alike.

If many matches nowadays seem to unfold in similar ways on different surfaces, the effect of increasing topspin probably bears more responsibility for it than any conspiracy to homogenize court surfaces. Blame the string manufacturers, not the court surface technicians or tournament directors.

Indoor Courts, Natural Surfaces, and Predictability

One issue that remains to be addressed is the widespread impression that indoor courts play faster than outdoor courts (and that even closing a retractable stadium roof has a dramatic effect on play). One obvious difference is that there’s no wind indoors. To some extent this can help aggressive players shorten points, since it allows them to aim closer to the lines and helps them serve more precisely (especially for players with high ball tosses). Yet a lack of wind can also help defenders extend points, by helping them anticipate the trajectory of their opponents’ shots and hit them cleanly. Other atmospheric parameters can also affect play indoors, in both directions, and I’ll address these in another post.

A more important factor in indoor events is the fact that typical indoor surfaces are not constructed in the same way as permanent outdoor surfaces, even if the visible material on top is identical. Very few indoor arenas are built exclusively for tennis—most spend more time hosting concerts, conferences, basketball games, hockey games, or other events than they do tennis tournaments. Thus the tennis playing surface must be laid on relatively thin, portable panels that can be removed and stored when not in use. Such a surface is unlikely to be as rigid as a permanent surface, and therefore is likely to generate lower bounces. Even more importantly, such temporary surfaces are unlikely to be perfectly flat, and will flex to different degrees in different places. This leads to less predictable bounces, which favor shorter points by drawing more frequent mis-hits and whiffed swings from defending players.

Although it’s very difficult to measure or model, predictability of a surface can be just as important a factor as friction and bounciness on outdoor surfaces as well. Made of non-homogeneous materials and varying particle sizes, sensitive to moisture levels and maintenance practices, and suffering significant wear during a single match, clay courts don’t play the same way in every spot where a ball might bounce. In some cases this may make them effectively play faster than measurements would indicate. With a living, reacting surface on a soil substrate subject to most of the same variations as clay, grass is the most unpredictable surface of all, and this undoubtedly contributes to its reputation as the sport’s fastest surface.

Final Note

I should give credit to Rod Cross, a physicist at the University of Sydney who has made an outstanding effort to model how balls bounce and many other phenomena in tennis, and whose work has been the best single guide in my own study of the subject. His model eliminates more simplifying assumptions and is therefore more sophisticated than mine. Unfortunately, the application of such advanced models is limited by the fact that many of the (few) measured values we have for µ and e for various court surfaces were measured by the ITF, and calculated using its own theoretical assumptions. Therefore, those assumptions are to some extent “cooked into” the measured values, and lead to inaccurate results when these values are fed into more sophisticated models.

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5 Responses to The Physics and Perception of Court Speed

  1. Nice article. How is slice affected compared to topspin?
    Also think that a lot of people’s perception of court speed is actually dependent on the mass / aerodynamic characteristics of the ball.

    • fogmount says:

      Slice shots react in a similar way to flat shots as they bounce, but because they land with backspin, they are even more resistant to the sliding-to-rolling transition, and can potentially be slowed even more, especially on high-friction surfaces. Such a loss of horizontal speed can have an especially dramatic effect with slices, since they must be hit slower than typical flat or topspin groundstrokes to prevent their aerodynamic lift from carrying them over the baseline before they bounce. Slices retain more speed when bouncing on low-friction surfaces, but this can work for or against the player who employs them, depending on the circumstances of the point. Drop shots, for example, can be more effective on high-friction surfaces (even high-bouncing ones) because the ball can lose all of its horizontal speed, staying so close to the net that an opponent can’t reach it.

      You are right to say that variations in the aerodynamic characteristics of the ball can have a significant effect on play—I’ll discuss that in an upcoming post. I’m not sure if the rules allow enough variation in mass to have a very significant effect—I’ll have to look into that.

  2. Arkon says:

    So you’re saying that the courts in general haven’t been slowed down significantly, even compared to 10 years ago? That most of the blame for the perceived “slowing down” of the game is on string manufacturers?

    The slow court speeds is even visible on the WTA, where players don’t hit with that much topspin to begin with. So I’m really not buying this whole “blame the string manufacturers, not the court manufacturers” argument.

    • fogmount says:

      We know courts at certain individual events have slowed down. There’s no way to know for sure whether courts in general have slowed down without measurements of their bounce characteristics, taken under similar conditions, now and in the past. As far as I know, such data doesn’t exist.

      I don’t have a simple answer to the question of how court speeds in general have changed between now and sometime in the past. What I’m saying is that even to an expert player, let alone a fan sitting in the stands or watching on TV, how fast or slow a court looks or feels depends on a lot of things besides the court itself. Comparing what’s “visible” in a match between today’s players playing today’s game with today’s equipment, compared to what past players did with old equipment, tells us very little about the court surface.

      Even in the WTA, more players are playing with more topspin than they did in previous generations.

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