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Follow The Bouncing Ball
Ball/Court Interaction - Part III

Jani Macari Pallis, 
Ph.D. Photo
Jani Macari Pallis, Ph.D.

During September, we examined how a ball's approach angle affects its outbound angle as spin rates vary on four different court surfaces (hard, green and red clay, grass). Last month we looked at rebound height and the horizontal distance these same balls attain. The four types of balls used were: Wilson US Open (hard court), Roland Garros (red clay), Slazenger Wimbledon (grass), and Wilson clay court.

This month we'll wrap things up and conclude this series by examining ball velocity before and after the bounce on these different surfaces as spin varies.

As a reminder, these were the pre-bounce spin rates used. There was quite a variation within each range, however these numbers can be used as a guide.
Flat : 0 rev/min
Low Topspin: 900 rev/min
Medium Topspin: 1500 rev/min
Heavy Topspin: 3000 rev/min
Medium Underspin: 1500 rev/min
Heavy Underspin: 2500 rev/min

In the charts below we define "Rebound Distance" as the ball's horizontal distance traveled to reach the ball's maximum rebound height. You can click on the thumbnail charts for full sized graphics. You might want to consider opening additional browser windows to view the larger versions of the chart. (NA means there was no data available in that category.)

Ball Speed Variation With Spin
We have placed the velocity after the bounce charts for all courts and balls next to the horizontal distance traveled to reach maximum height. Even without expanding the charts its easy to see that velocity out and the distance traveled after the bounce charts look very similar. You can see a strong correlation between the two parameters. This makes sense since you know that the ball travels much farther horizontally than it does vertically for serves and most ground strokes. As you will see, typically the horizontal component of velocity dominates the total velocity of a tennis ball.

What do we observe? The more topspin on the ball the less the ball speed is reduced after the bounce. For underspin, aside from the grass court, the more underspin on the ball before the bounce, the greater the reduction in speed. Why the difference? Remember after the bounce just about all balls have topspin on them. A ball hit with underspin is going to change spin direction into topspin after the bounce. Changing direction slows the ball down.

Velocity Out - Distance

Graph Graph

Court: Ball: Velocity Before
Bounce (MPH):
Velocity After
Bounce (MPH):
Green Clay Wilson US Open33 20 39%
Green Clay Wilson Clay 34 21 38%
Red Clay Wilson US Open 29 1838%
Red Clay Roland Garros31 1939%
Hard US Open 32 20 38%
Grass US Open 35 18 49%
Grass Wimbledon 35 18 49%

Low Topspin
Velocity Out - Distance

Graph Graph

Court: Ball: Velocity Before
Bounce (MPH):
Velocity After
Bounce (MPH):
Green Clay Wilson US Open34 23 32%
Green Clay Wilson Clay 33 23 30%
Red Clay Wilson US Open 31 2132%
Red Clay Roland Garros32 21 34%
Hard US Open 35 24 31%
Grass US Open 38 21 45%
Grass Wimbledon 37 22 41%

Medium Topspin
Velocity Out - Distance

Graph Graph

Court: Ball: Velocity Before
Bounce (MPH):
Velocity After
Bounce (MPH):
Green Clay Wilson US Open-NA- -NA--NA-
Green Clay Wilson Clay 36 27 25%
Red Clay Wilson US Open 35 2626%
Red Clay Roland Garros-NA- -NA--NA-
Hard US Open 37 28 24%
Grass US Open 39 24 38%
Grass Wimbledon 39 26 33%

Heavy Topspin
Velocity Out - Distance

Graph Graph

Court: Ball: Velocity Before
Bounce (MPH):
Velocity After
Bounce (MPH):
Green Clay Wilson US Open37 31 16%
Green Clay Wilson Clay 37 32 14%
Red Clay Wilson US Open 36 3017%
Red Clay Roland Garros35 2917%
Hard US Open 35 31 11%
Grass US Open 39 28 28%
Grass Wimbledon 41 30 27%

Medium Underspin
Velocity Out - Distance

Graph Graph

Court: Ball: Velocity Before
Bounce (MPH):
Velocity After
Bounce (MPH):
Green Clay Wilson US Open32 17 47%
Green Clay Wilson Clay 31 16 48%
Red Clay Wilson US Open 31 1745%
Red Clay Roland Garros28 1546%
Hard US Open 29 15 48%
Grass US Open 36 18 50%
Grass Wimbledon 35 17 51%

Heavy Underspin
Velocity Out - Distance

Graph Graph

Court: Ball: Velocity Before
Bounce (MPH):
Velocity After
Bounce (MPH):
Green Clay Wilson US Open23 13 43%
Green Clay Wilson Clay 33 19 42%
Red Clay Wilson US Open 31 1842%
Red Clay Roland Garros30 1743%
Hard US Open 28 16 43%
Grass US Open 33 16 52%
Grass Wimbledon 33 16 52%

The velocity charts were very interesting! Below we've summarized the differences in velocity reduction after the bounce for the "slow" (red clay, green clay and hard courts), versus the "fast" (grass court). The "slow" average is the average of those 3 courts.

Keep in mind that a low velocity reduction means that the velocity after the bounce remained closer to the value of the velocity before the bounce. Compare the flat average reduction to the topspin reductions. As we add more topspin, the balls' velocity after the bounce is not reduced as much. You can see the velocity reduction numbers drop from low topspin to heavy topspin.

Compare the medium underspin and heavy underspin to the flat and topspin balls. Medium underspin in our study reduced the ball's velocity the most.

Look at the reductions for the grass court in the first chart! The results show the following: grass courts reduce the velocity of the ball more than the other courts. How can this be? Grass is supposed to be the "fast" court! Note also that in all cases the grass court velocity in was higher or equal to the other court/ball's velocity in. So what does "fast" mean!

Spin Type Average Velocity
Reduction for "Slow" Courts
Red, Green, Hard:
Average Velocity
Reduction for "Fast" Court
Flat38.4 49.0
Low Topspin 31.8 43.0
Medium Topspin 25.0 35.5
Heavy Topspin 15.027.5
Medium Underspin 46.8 50.5
Heavy Underspin 42.6 52.0

Ball/Court Interaction
What Does "Fast" Mean?

I'll be the first to admit grass courts are a challenge to conduct studies on - just ask any of the folks doing ball/green interaction in golf. After all the grass is "living", it has moisture on it and in it which changes throughout the day. Blades of grass change direction with the sun and with shade. Having seen the pristine manicured courts at Wimbledon, I can tell you the courts we used - well, didn't look like those at Wimbledon. With this in mind, I can say that these are the results our team obtained; under different environmental conditions and on a differently maintained court different results might be seen. On the other hand, other researchers studying typical grass courts (versus Wimbledon) might find this information useful as a comparison.

In Part I we discussed the effects we expected to see. According to Professor Howard Brody's book, Tennis Science for Tennis Players, interaction between the ball and the court:

  1. causes the angle out to change from the angle in;
  2. the smaller the friction the smaller the rebound angle;
  3. the smaller the friction the faster the court;
  4. the larger the friction the greater the rebound angle;
  5. the greater the friction the slower the court.

We know that the coefficient of restitution (parameter which deals with ball rebound and bounce) affects the vertical velocity and the coefficient of friction (parameter which deals with the friction effects of the ball rubbing against the court) affects the horizontal velocity. We call the horizontal velocity Vx and vertical velocity Vy. The velocity we have been reporting in our charts has been the total velocity V. We captured Vx and Vy from the video footage and calculated V using the following equation: V=Vx2 + Vy2)1/2. We also used Vx and Vy to get our rebound angle. The rebound angle is a ratio of Vy to Vx.


Prof. Brody's book tells us that more friction means a smaller rebound angle and a faster court. Clearly, we saw much smaller rebound angles on the grass court. What makes a small rebound angle? Look at the diagram below.


You can see that the smaller the rebound angle the smaller the vertical velocity or the larger the horizontal velocity. A small vertical velocity means a small coefficient of restitution - the ball isn't going to bounce high; a larger horizontal velocity component signifies low friction, a small rebound angle. The large horizontal component of velocity means one more thing - the ball will reach you faster. The diagram below demonstrates this.


In the diagram the different colored balls represent the same point in time. (The red balls are all at the same point in time but not in space; the blue balls are all at the same point in time but not in space , etc.) Let's assume that the balls both have the same velocity when they first bounce off of the court. Court 1 acts like the grass court - its trajectory is a straighter line. Court 2 acts like one of the slower courts - its trajectory is more of a curve. You can see that the red ball has traveled farther on Court 1 in the same amount of time. If you are standing 7 feet away from the ball you will have less time to react to get to the ball.

Another short analysis was conducted to prove this. Four of the original (not average) test cases were selected; one for each court type using the US Open ball. In each of these test cases the velocity after the bounce was roughly 19 miles/hour. We examined which ball would reach a player standing at the same position on the different courts, first. We arbitrarily selected 7 feet from the bounce.

Green Clay - Grass


Hard - Red Clay


The results are clear, the grass court ball not only reached the player first, but it had a faster velocity when it reached the player. This was because its horizontal velocity Vx was faster than the balls from the other courts.

There are some very interesting things to note in these charts - and they are fun to play around with. When you open the charts up you will see columns for time, Vx, Vy, V and the distance from the bounce in inches.

Seven feet is 84 inches. Go down the distance column for the grass court until you reach 84 inches. (Actually, there is one entry at 83.1 inches and the next entry at 85.3 inches.) Look at the time and velocities: the time is roughly .3 seconds, the horizontal velocity is 15.66 miles/hour, the total velocity is about 15.70 miles/hour.

Look at the data for the green court. The ball doesn't reach 84 inches until the time is .337, the horizontal velocity is 13.6 and the total velocity is 14.08. That's almost 2 miles/hour less.

The grass ball reaches a player sooner. You have less time to react and that could qualify this grass court as "faster".

If you look farther down the grass court court at 98.4 inches from the bounce, you will notice something else, the vertical velocity becomes negative. This means the ball is now dropping. If you look at the green clay court the ball is still climbing. The grass ball starts dropping to the ground sooner than the balls off of the other court - a player has less time to react for this reason as well.

Summary of Results
Finally, just to put it all together and into perspective, here is a summary of all of the results from all three parts of this article: velocity and angle before and after the bounce, rebound height, horizontal (down the court) distance, for all four ball types and court surfaces under the defined spin rates.

Ball/Spin/Court Velocity In Velocity Out Angle In Angle Out Rebound Height Horizontal Distance
Grass - US Open Velocity In Velocity Out Angle In Angle OutRebound Height Horizontal Distance
Flat 35 18 25 292687
Low Top 38 21 25 2933114
Medium Top 39 24 23 2330128
High Top 39 28 25 1926144
Medium Under 36 18 22 242083
Heavy Under 33 16 23 292377
Grass - Wimbleton Velocity In Velocity Out Angle In Angle OutRebound Height Horizontal Distance
Flat 35 18 24 292894
Low Top 37 22 24 2632123
Medium Top 39 26 23 2231140
High Top 41 30 23 1725148
Medium Under 35 17 23 262179
Heavy Under 33 16 23 261972
Green - US Open Velocity In Velocity Out Angle In Angle OutRebound Height Horizontal Distance
Flat 33 20 27 3851122
Low Top 34 23 28 3658148
Medium Top -- -- -- ------
High Top 37 31 28 2861209
Medium Under 32 17 25 404498
Heavy Under 23 13 21 312472
Green - Wilson Clay Velocity In Velocity Out Angle In Angle OutRebound Height Horizontal Distance
Flat 34 21 28 4160126
Low Top 33 23 29 3658143
Medium Top 36 27 25 3058186
High Top 37 32 29 2966213
Medium Under 31 16 26 424492
Heavy Under 33 19 21 3134104
Red - US Open Velocity In Velocity Out Angle In Angle OutRebound Height Horizontal Distance
Flat 29 18 27 3845104
Low Top 31 21 26 3551130
Medium Top 35 26 23 2949146
High Top 36 30 25 2549186
Medium Under 31 17 24 394195
Heavy Under 31 18 21 313199
Red - Roland Garros Velocity In Velocity Out Angle In Angle OutRebound Height Horizontal Distance
Flat 31 19 25 3645109
Low Top 32 21 26 3349136
Medium Top -- -- -- ------
High Top 35 29 26 2751176
Medium Under 28 15 25 393778
Heavy Under 30 17 23 343395
Hard Court - US Open Velocity In Velocity Out Angle In Angle OutRebound Height Horizontal Distance
Flat 32 20 24 3545116
Low Top 35 24 27 3357163
Medium Top 37 28 22 2853184
High Top 35 31 26 2553204
Medium Under 29 15 25 423981
Heavy Under 28 16 21 312681

The Latest Technology
Although the objective of this work was to summarize these parameters for educational purposes and to give players and coaches a feel for variations, these types of tests are also conducted to determine court pace (court speed).

Our tests were conducted using high speed video cameras to capture the ball's behavior with computer software used to calculate the different parameter results at a later time. (I can tell you this was tough on all of our eyes during the analysis phase.)

Today, a compact measuring device is used which is able to calculate ball size (radius), the angle and ball velocity before and after the bounce, contact time with the surface, and how far the ball slides when in contact with the surface. The Wassing sestée has been used by the International Tennis Federation, Lawn Tennis Association (the governing body for the game of tennis in Great Britain) and the French Tennis Federation.

Future Article On Shoes
I'm beginning to prepare an article on tennis shoes. If you have shoe technology or shoe/court interaction questions please feel free to send them to me using this form and I will try and incorporate your questions into the article.

Until Next Month ... Jani

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This column is copyrighted by Jani Macari Pallis, Ph.D., all rights reserved.

Dr. Jani Macari Pallis is the founder and CEO of Cislunar Aerospace, Inc., an engineering and research firm in San Francisco. In addition to her engineering practice, she has led two collaborations between NASA and Cislunar, creating educational materials on the aerodynamics of sports for pre-college students and educators. As the head of NASA's "Aerodynamics in Sports" project, she has led a team of researchers investigating the aerodynamics, physics and biomechanics of tennis. The group has conducted high speed video data capture at the US Open and research of ball/court interaction, footwork, serve speeds, trajectories and ball aerodynamics. Pallis received a BS and MS from the Georgia Institute of Technology, an MS in mechanical engineering from the University of California, Berkeley and a Ph.D. in mechanical and aeronautical engineering from the University of California, Davis. She is a member of the Executive Committee of The International Sports Engineering Association.

Questions and comments about these columns can be directed to Jani by using this form.


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