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Follow the Bouncing Ball

Understanding how the ball bounces can help you and your students pick and read shots.

By Crawford Lindsey

Watching tennis played on the red clay of Roland Garros at the French Open and on the Centre Court grass at Wimbledon can seem like watching two different sports. The events, flow, and look of the game are completely different. And for good reason — the bounce of the ball is completely different on each surface, and it is the bounce that determines the game.

That five-millisecond bounce dictates everything, including shot selection, tactics, strategy, stroke mechanics, grips, and training. During that five one-thousandths of a second, the ball’s speed, spin, direction, height, and angle are changed. And these in turn dictate what kind of strokes and strategy a player adopts as he or she learns the game.

When a ball hits the court, a vertical force pushes the ball up and a horizontal force acts to slow the ball and change its spin. These two forces have far-reaching consequences, with effects that may seem contrary to intuition and experience.

VERTICAL BOUNCE

Vertical Bounce Factors

Vertical speed and court hardness. The magnitude of the upward force, known as the “ground reaction force,” is determined by the vertical incident speed of the ball and the relative hardness of the court and the ball. The harder the court, the higher the ball will bounce. If the court is soft, energy will be lost deforming the court surface. The surface does not spring back fast enough or efficiently enough to aid the ball in its bounce, so that energy is “lost.” (This is the opposite of what happens on strings. The softer the strings, the faster and higher the ball bounces. This is due to both the greater resiliency of strings and the fact that the ball deforms less on soft strings.)

Stages of Bouncing (some or all may occur)
Flying
Landing
Slowing
Spinning
Skidding
Rolling
Biting
Stretching
Hopping
Flying

Comparing vertical speeds. The ratio of the ball’s vertical speed after the bounce to that before is known as the “coefficient of restitution” (COR). If the vertical speed after the bounce is faster on one court versus another, the ball will bounce higher on that court. The COR is about 0.6 for grass, 0.83 for hard courts, and 0.85 for clay courts. That means that a ball’s vertical bounce will be highest and fastest on clay, lowest and slowest on grass.

Limits on Vertical Bounce

Ball deformation. The vertical bounce speed depends on the ball’s vertical speed before it hits the court. For a given court, the faster the ball hits the surface, the faster and higher it will bounce. The efficiency of the bounce will deteriorate, however. Higher impact speeds will cause the ratio of post- to pre-impact velocities (COR) to go down. That is because the ball will deform more with faster impacts. The more it deforms, the more the energy used in that deformation becomes unusable for ball bounce.

Ball stiffness. In general, at moderate speeds, about 45 percent of the energy that goes into ball deformation is lost. This percentage goes up at higher impact speeds. But it does not go up a great deal. That’s because the ball also gets stiffer the more it is deformed. The gain in stiffness tends to decrease the amount of deformation that occurs for each unit of increase in impact speed. In general, a faster ball will always bounce higher than a slower one, but it will not bounce higher in direct proportion to the speed increase, but something less.

Factors Influencing Vertical Speed

Trajectory. What will be the vertical speed of the ball when it impacts the court just prior to the bounce? This is important for both the vertical and horizontal bounce of the ball (see below). On most shots (all except volleys, overheads and some serves in which you hit down into the court), the vertical speed of the ball toward the court is almost totally determined by gravity. Simply put, the ball drops from the maximum height attained in its trajectory. You don’t usually hit the ball down onto the court; it falls into the court. The horizontal speed of the ball before the bounce has nothing to do with the vertical impact speed or how high it will bounce.

Spin. The spin of the ball will also affect the vertical impact speed. Topspin creates an airflow force known as the Magnus force. This force pushes a topspin ball down and pushes a backspin ball upward. Furthermore, a ball hit with topspin is likely to reach a higher apex in its trajectory, so it will drop from a greater height, gain more speed, and hit at a steeper angle. Topspin balls will therefore bounce faster in the vertical direction and higher than will balls hit flat or with backspin.

HORIZONTAL BOUNCE

Friction

What is friction? All the real interesting stuff happens as a result of the friction force. Friction determines the angle of bounce, the horizontal bounce speed, and the spin. It does not affect the vertical bounce, however. Friction acts parallel to and opposite the direction of motion of the ball on the surface. Friction arises whenever a force is applied to move one object across the surface of another.

Factors in Ball Bounce
Hardness of court
Hardness of ball
Vertical speed
Force pushing ball up
Force pushing ball down
Spin speed and direction before bounce
Horizontal speed before bounce
Relative roughness of court and ball
Magnitude, time, and direction of friction

There are three types of friction, all of which can play a role in the bounce of the ball from the court. The three types are sliding, rolling, and static friction. Unless we are referring to a specific type, we will use the generic term “friction.”

Determinants of friction. Contrary to intuition, the magnitude of the friction force does not depend on the surface areas in contact or the horizontal speed of the ball. It depends on two things — the force pushing the ball and court together (called the “normal” force, which is the opposite of the ground reaction force pushing the ball back off the court), and the relative roughness of the ball and the court. The latter is quantified by the coefficient of friction (COF).

Comparing court friction. Every court surface has a different COF with respect to a tennis ball. COF is usually a number between 0 and 1 but numbers greater than 1 are also possible. Grass is about 0.6, hard courts about 0.7, and clay about 0.8. The higher the COF, the more friction will be generated and the slower will be the resulting horizontal speed of the ball.

Surprisingly, clay has both the highest COR and COF and grass the lowest (Table 1).

Consequently, a surface with a fast and high vertical bounce will also bounce slow and steep horizontally. And a surface that has a low and slow vertical bounce will tend to bounce fast at a low angle horizontally. That means fast courts are usually slow in the vertical direction and slow courts are usually fast in the vertical direction.

Trajectory and friction. The impacting ball’s angle of incidence is important by virtue of what it implies. By itself, it does not determine the vertical impact speed, but rather indicates what the flight of the ball probably looked like.

Table 1

Court	COR (speed out/speed in)	Bounce Height (% of trajectory height)	COF
Grass	.6	36%	.6
Hard	.83	69%	.7
Clay	.85	72%	.8

Imagine that two balls are hit with the same speed from the baseline to land at the same location at the opposite service line. If one is launched at 30 degrees and the other at 60 degrees, then the 60-degree ball will follow a steeper, higher trajectory and thus gain more vertical speed on the way down. As a result, it will impact at a steeper angle and experience a greater friction force.

You would think that if there is more friction, then the more vertically impacting ball will also slow down the most (as a percentage of its horizontal speed). This is not necessarily true because the combination of a slower horizontal speed and more friction can magically combine to reduce the time the friction force acts. This is, in turn, results in less slowing of the ball.

That’s where the plot thickens, as we will shortly see.

Slowing and Spinning

Speed and Bounce Angle. The angle of the bounce is determined by the ratio of the vertical speed to the horizontal speed. If the vertical speed is greater than the horizontal, the ball will bounce greater than a 45-degree angle. If it is less, it will bounce less than a 45-degree angle. The more the ball slows during the bounce, the steeper will be the bounce angle. No matter the angle of impact, balls with the same vertical velocity will bounce to the same height in the same time on the same court. The steepness of this “climb” will depend on the horizontal velocity.

Spin Change. When the ball first hits the court, it is squashed and it slides and skids across the surface. Friction immediately begins to slow the horizontal speed by pushing backward on the bottom of the ball. This same action exerts a torque on the ball around the ball’s center. This causes the ball to spin in the direction of the force, resulting in topspin.

Incident Spin Affects Speed and Spin. The big question is how much will the ball slow down and how much will it spin. The answer is that it depends. It depends on the magnitude of the friction force, which we have looked at, and also on the length of time and the direction in which it operates. And these depend on the direction and rate of spin just before the ball touches the court. The direction of the spin refers to the direction at the bottom of the ball in relation to the motion of the ball as a whole.

The speed of the bottom of the ball is thus the speed of the ball’s forward motion plus or minus the ball’s spin speed. For a ball hit with backspin, the bottom of the ball is traveling faster than the entire ball. For a ball hit with topspin, the bottom is traveling slower than the rest of the ball. If the ball has no spin, it will instantly gain topspin. If the backward spin speed is the same as the ball’s forward speed, then the speed at the bottom of the ball will be zero. The forward motion and backward spin cancel each other, and the bottom of the ball is actually momentarily at rest. This is an important event, as we will see.

Skidding

Whenever speed and spin do not cancel in this manner (balls incident less than 20 degrees will skid the entire bounce), the ball will skid across the court surface. Throughout the skid, friction will continue to slow the forward speed and increase the topspin. When the forward speed slows enough and the topspin becomes fast enough, the two cancel, the bottom of the ball (not the entire ball) is at rest on the court surface, skidding stops, and sliding friction disappears, even though the bounce is not yet finished. When the speed and spin are so aligned, the ball stops skidding and begins to roll. This is an important transition because the magnitude of rolling friction is much less than sliding friction, and for our purposes can be considered to be zero.

From Skidding to Rolling

The reason there is no friction during rolling is that the bottom of the ball is at rest, and thus there is no movement of one surface against the other, nor is there the sliding friction force that arises with such movement. When sliding friction stops, so too does the horizontal deceleration and rotational acceleration.

The phenomenon of motion occurring even though the surfaces in contact are at rest is a familiar situation. When a bicycle wheel rolls, the forward speed of the bike is the same as the spin speed of the wheels, so the bottom of the wheel is always at rest on the road. The case is similar with walking. When you put your foot down, your foot remains at rest on the ground as you propel your body forward over that foot. No matter how fast you run, your foot must come to a complete stop on each step.

Spin

How long will friction last? This canceling of forward motion and backward spin manifests itself a little differently in backspin and topspin shots. If you hit the ball with a lot of backspin, it will take longer for friction to reverse the spin direction and accelerate the spin up to the opposite of the horizontal speed. During that time, friction will also be slowing the ball.

Figure 1. A backspin ball will slow horizontally more than a topspin ball hitting at the same angle and speed. Usually, and depending on the angle of incidence, the horizontal speed will slow more than the vertical speed. A backspin ball will therefore bounce at a steeper angle than the incident angle. The topspin ball is more likely to reach a rolling or biting condition where there is no friction and therefore will slow less horizontally than the backspin ball. Depending on the incident angle, the horizontal speed will slow less than the vertical bounce speed. Consequently, a topspin ball is most likely to bounce at a shallower angle than the incident angle.

However, if you hit the ball with heavy topspin, the spin is already going in the opposite direction from the forward trajectory, and if you hit with very heavy topspin, it might already be spinning almost as fast as the forward speed. If this is the case, friction does not have to work for very long to slow the forward speed and increase the spin speed until they are equal. In other words, a topspin ball will slow less than a backspin ball and it will change its spin less because friction works on it for a shorter time.

Topspin bounces lower than backspin in the lab. What are the consequences of this? It means that for two balls hitting the court at the same angle and speed, but one with topspin and the other with backspin, the topspin ball will come upon you faster and lower than a backspin ball. Lower, because it travels farther horizontally for each unit of vertical bounce than does the backspin ball (see Figure 1).

Backspin bounces lower than topspin on the court. But wait. This is completely opposite of what you experience on court. You know that the backspin ball will stay lower and seem to come to you faster. The answer is that topspin and backspin balls rarely will hit the court at the same angle. For a backspin and topspin groundstroke from the baseline to hit the same spot at the same angle would be virtually impossible. You could get them to hit the same spot, but the trajectories would be radically different. The topspin shot would land at a much steeper angle. A cannon can fire topspin and backspin at the same angle at the court, but it won’t happen while you are playing.

But nonetheless, our usual perception is wrong. We can’t say that backspin causes the ball to bounce lower and faster than topspin. Instead we must realize that backspin causes the ball to slow more and bounce steeper than topspin, but that backspin almost always takes lower trajectories and therefore lower bounces. You’re never comparing apples to apples in realistic court-playing situations.

From Rolling To Biting

We said the ball will start to roll when the spin speed and the horizontal speed are equal but opposite and that friction stops working at that point. The sooner in the bounce that happens, the less the ball will slow and change spin speed. We find however that the rolling transition is really more of a theoretical transition point.

As soon as the bottom of the ball comes to rest, the friction goes to zero, as if it were rolling, but instantly goes past zero into negative territory. In other words, it starts acting in the opposite direction. This would not happen with a hard object like a bowling ball. In fact, you can easily observe the transition from sliding to rolling as a bowling ball goes down the lane. At that point there are no other forces (ignoring air resistance) acting on the bowling ball.

A tennis ball is different. When the bottom of the ball is at rest and while the ball is squashed on the court, the bottom of the ball “grips” or “bites” the court. The bottom actually gets stuck, as if it were glued to the court. The remainder of the ball is still rotating and translating faster than the bottom, and the ball therefore stretches horizontally. Its backside, behind the “stuck” point at the bottom of the ball, goes into tension, and the front side goes into compression as the faster pieces of ball start piling up in front of the stuck point.

From Biting To Stretching

We now have an elastic force acting on the ball that a bowling ball would not experience. The elastic force is trying to pull the back end and push the front end of the contact point across the surface in the backward direction. Consequently the friction between the court and ball pushes back in the forward direction to resist this elastic force. Since the bottom of the ball still is not moving, we say that it is static friction that is resisting the elastic force. It is a consequence of the force pushing the two surfaces together. It is during this time, which may exist for less than a millisecond, that we say the ball is “biting” the court.

From Stretching To Hopping

But this is fleeting, and the elastic force may overcome the static friction force before the ball finishes its bounce. The bottom of the ball then begins to skid backwards, because now it is rotating faster backwards than it is moving forward. So, sliding friction takes over from static friction and acts to slow the spin down and speed the ball up. This is just the opposite of what happened when the ball first hit the court. And it is now that bizarre things can happen.

Depending on when the biting occurs during the bounce (i.e., sooner or later), you will have a longer or shorter time during the bounce when the friction force is actually propelling the ball forward. There are even situations where the ball can bounce at a greater horizontal speed than it landed. We have all seen bounces that seemed to accelerate and hop after the bounce. These same balls will also have less topspin than you would expect, since the same force is slowing the topspin.

Topspin vs Backspin Bounce (incident at same angle and speed)	Backspin vs Topspin Bounce (incident at same angle and speed)
Bounces same speed and height vertically	Bounces same speed and height vertically
Faster horizontal speed	Slower horizontal speed
More topspin	Less topspin
Bounces shallower than incident angle	Bounces steeper than incident angle

Balls that hit the court at less than 20 degrees will slide throughout the bounce and sliding friction will be evident throughout the bounce. Above 20 degrees, balls will bite the court at some point. When they do, friction will reverse itself. The steeper the angle of incidence, the sooner that is likely to happen in the bounce.

Conclusion

We have learned some bizarre things in this little excursion into bouncing balls. First, we have learned that for balls incident at the same angle and speed, topspin bounces faster and lower than backspin — just the opposite of what we thought. But we also saw that topspin and backspin balls don’t ever land at the same angle and speed in normal playing conditions. Given the way they do land (backspin with lower trajectories), our perception that backspin bounces lower and faster than topspin is correct, but only because they start out that way.

We also learned that the relationship of incident spin speed and direction to horizontal speed determines how long and in what direction friction will act and what its results will be. All this happens in 5 to 7 milliseconds. Who would have thought that so much could happen before our eyes in such a short time with such game-critical consequences? Better think again!

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Editor’s note: Much more on bounce can be found in The Physics and Technology of Tennis, by Howard Brody, Rod Cross, and Crawford Lindsey — available from the USRSA. Many of the findings reported in this article are based on research by Brody and Cross.

See all articles by Crawford Lindsey

About the Author

Crawford Lindsey  is co-author of The Physics and Technology of Tennis and Technical Tennis

From the April 2004 issue
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