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Raw Racquet Power

By Rod Cross

When Crawford Lindsey and I were writing Technical Tennis, our biggest single problem was to find a meaningful word or phrase to replace “Apparent Coefficient of Restitution,” or ACOR. This is the technically correct phrase used to describe how well a ball bounces off a stationary racquet. The average tennis player would probably tune out if we used that phrase.

We eventually decided to call it “Rebound Power,” or RP, which was suggested by Howard Brody. In any case, catchy name or not, why would anyone care how well a ball bounces off a racquet that is not being swung at the ball? The answer is that the power built into the racquet needs to be carefully distinguished from the power that a player can get out of it.

When a player talks about racquet power, he usually has in mind the power that is built into the racquet by the manufacturer using whatever technical tricks it can come up with, such as an improved string suspension system or some clever way of constructing the frame. When a manufacturer designs a new racquet, it might take an existing frame, modify it in some way, and end up with a racquet that it hope will be a better, perhaps more powerful version of the original. How would one test the racquet to see if it is more powerful or not? The obvious way would be to serve a ball at a fixed racquet speed and use a radar gun to see if the ball is served any faster with the modified racquet. If it is, then the modified racquet is indeed more powerful.

A simpler test is to fire a ball at a stationary, freely-suspended or hand-held racquet and see if the ball bounces off the racquet any faster. This test is simpler because it is easier to measure the speed of a ball than to measure the speed of a racquet. A problem with measuring racquet speed is that different parts of the racquet travel at different speeds. Another problem is ensuring that the racquet is swung at the same speed each time. In fact, both tests are entirely equivalent and yield the same answer. In a stationary racquet test, one measures the speed of the incoming ball and the speed of the outgoing ball. The ratio of these two speeds (speed out/speed in) is rebound power (RP). It is a number that doesn’t involve the player or his ability to swing the racquet. The RP is a measure of the power built into the racquet. In general, heavy racquets have a bigger RP than light racquets, so an increase in RP without an increase in racquet weight is an indication that the manufacturer has come up with an improved, more powerful design.

Determining Rebound Power

Crawford had the hopeful idea that it would be nice to measure the RP of every performance racquet on the market and publish the results each year in RSI. The problem is that it takes several hours to measure the RP properly, taking the average of perhaps 5 or 10 bounces at each of several different spots on the strings. It is a lot simpler to calculate the RP using other measured properties of a racquet. The RP can be calculated quite accurately for an impact in the middle of the strings, but RP calculations are less accurate at other impact points because it is difficult to estimate energy losses due to frame vibrations. This is not a problem in the middle of the strings because the frame doesn’t vibrate at all for such an impact. I measured the RP in the middle of the strings for five radically different racquets and got the same answers as the calculated values, demonstrating that it is easier to calculate the RP than measuring it and the answer will be the same, at least for impacts near the middle of the strings.

Figure 1: Calculated values of RP at a point 16 cm from the tip of the racquet, vs. racquet weight, for 268 different racquets.

Racquet power vs. Racquet weight

Using these formulas (which can be found in The Physics and Technology of Tennis), I calculated the RP for an impact 16 cm from the tip of the 268 different racquets listed in the March 2004 and March 2005 issues of RSI. The results are shown in Figures 1 and 2. Figure 1 shows a graph of RP vs. racquet weight. The points on the graph are scattered all over the place, but it is clear that the RP for heavy racquets is generally larger than the RP for light racquets, as expected. A tennis ball bounces higher off a heavy slab of concrete than a light sheet of paper. The RP for a concrete slab is about 0.75, as specified by the rules of tennis. The RP for a hand-held racquet, when the ball impacts the middle of the strings, varies from about 0.31 to about 0.43. In Figure 1 the points are scattered all over the place because the RP depends mainly on the weight of the head rather than the weight of the whole racquet. Two racquets of the same weight will have different values of RP if one is head- heavy and the other is head light. The head heavy racquet will have a larger RP, so it has more inbuilt power. If both racquets are swung at the same speed, the ball will come off the head-heavy racquet faster.

Amazing Results

Figure 2 shows RP vs. swingweight for all racquets. The result is simply amazing. Instead of having the 268 dots scattered all over the place, the dots line up perfectly along four different curved lines. The four curves correspond to different racquet lengths. All racquets of the same length lie on the same curve, with short racquets having a bigger RP than long racquets. The result in Figure 2 shows that any two racquets of the same length and the same swingweight will have exactly the same RP, regardless of their weights and regardless of their balance points. The inbuilt power of a racquet in the middle of the strings therefore depends only on the length and swingweight of the racquet, and on nothing else.

Figure 2: Calculated values of RP at a point 16 cm from the tip of the racquet, vs. swingweight, for the same racquets as those in Fig. 1.

Rebound power vs. Swingweight

There is a simple reason that long racquets have a smaller RP. In order for a long racquet to have the same swingweight as a short racquet, weight has to move out of the head and relocated closer to the handle. Since RP is determined mainly by weight in the head, a long racquet must therefore have a smaller RP than a short racquet (at any given swingweight).

The inbuilt power can be increased slightly by reducing string tension, but it was assumed in Figures 1 and 2 that all racquets had the same stringbed stiffness when calculating the RP. In other words, the string tension was adjusted for every racquet to compensate for differences in head size and string pattern so that the stringbed stiffness would be the same. This was not done on an individual basis. Rather, it was assumed in the calculations.

The RP for impact points away from the middle of the strings might tell a different story, but it won’t be a radically different story. If a racquet has a large RP in the middle of the strings, then in principle it should be large everywhere else over the stringbed. However, it won’t be proportionally larger because there are other factors to consider when a ball impacts away from the middle of the strings. Impacts near the tip and throat result in frame vibrations that act to reduce the RP compared with that of an infinitely stiff racquet. Consequently, if two racquets have the same length and swingweight, then the stiffer of the two will have a higher RP near the tip and the throat (but they will have the same RP in the middle of the strings).

Impacts away from the long axis, near the edge of the frame at the 3 and 9 o’clock positions, cause the racquet to twist around the long axis, which also results in a lowering of the RP. In this case, if two racquets have the same length and swingweight, then the one with the higher twistweight will be the most powerful on impacts towards the sides of the frame. There is plenty that a manufacturer can do, apart from increasing swingweight, to increase the inbuilt power of a racquet outside of the middle of the strings. Whether or not the result is a racquet that feels good in the hands of a particular player is a completely different story. There is no way that I can calculate the feel of a racquet. It just doesn’t compute.

See all articles by Rod Cross

About the Author

Rod Cross retired in 2003 as an honorary member of the Sydney University staff and continues to work on the physics of sport and forensic physics. He is a physicist and co-author of The Physics and Technology of Tennis available from the USRSA.

From the February 2006 issue
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