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!

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 V_x and vertical velocity V_y. The velocity we have been reporting in our charts has been the total velocity V. We captured V_x and V_y from the video footage and calculated V using the following equation: V=V_x² + V_y²)^1/2. We also used V_x and V_y to get our rebound angle. The rebound angle is a ratio of V_y to V_x.

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.

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 V_x 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, V_x, V_y, 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 Out	Rebound Height	Horizontal Distance
Flat	35	18	25	29	26	87
Low Top	38	21	25	29	33	114
Medium Top	39	24	23	23	30	128
High Top	39	28	25	19	26	144
Medium Under	36	18	22	24	20	83
Heavy Under	33	16	23	29	23	77
Grass - Wimbleton	Velocity In	Velocity Out	Angle In	Angle Out	Rebound Height	Horizontal Distance
Flat	35	18	24	29	28	94
Low Top	37	22	24	26	32	123
Medium Top	39	26	23	22	31	140
High Top	41	30	23	17	25	148
Medium Under	35	17	23	26	21	79
Heavy Under	33	16	23	26	19	72
Green - US Open	Velocity In	Velocity Out	Angle In	Angle Out	Rebound Height	Horizontal Distance
Flat	33	20	27	38	51	122
Low Top	34	23	28	36	58	148
Medium Top	--	--	--	--	--	--
High Top	37	31	28	28	61	209
Medium Under	32	17	25	40	44	98
Heavy Under	23	13	21	31	24	72
Green - Wilson Clay	Velocity In	Velocity Out	Angle In	Angle Out	Rebound Height	Horizontal Distance
Flat	34	21	28	41	60	126
Low Top	33	23	29	36	58	143
Medium Top	36	27	25	30	58	186
High Top	37	32	29	29	66	213
Medium Under	31	16	26	42	44	92
Heavy Under	33	19	21	31	34	104
Red - US Open	Velocity In	Velocity Out	Angle In	Angle Out	Rebound Height	Horizontal Distance
Flat	29	18	27	38	45	104
Low Top	31	21	26	35	51	130
Medium Top	35	26	23	29	49	146
High Top	36	30	25	25	49	186
Medium Under	31	17	24	39	41	95
Heavy Under	31	18	21	31	31	99
Red - Roland Garros	Velocity In	Velocity Out	Angle In	Angle Out	Rebound Height	Horizontal Distance
Flat	31	19	25	36	45	109
Low Top	32	21	26	33	49	136
Medium Top	--	--	--	--	--	--
High Top	35	29	26	27	51	176
Medium Under	28	15	25	39	37	78
Heavy Under	30	17	23	34	33	95
Hard Court - US Open	Velocity In	Velocity Out	Angle In	Angle Out	Rebound Height	Horizontal Distance
Flat	32	20	24	35	45	116
Low Top	35	24	27	33	57	163
Medium Top	37	28	22	28	53	184
High Top	35	31	26	25	53	204
Medium Under	29	15	25	42	39	81
Heavy Under	28	16	21	31	26	81

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