Excerpts from Tennis Science for Tennis Players by Howard Brody


Excerpts from Tennis Science for Tennis Players by Howard Brody, 1987, U of Penn.
Still available new (or used) in paperback from Amazon.com via the link above.

    Tennis Science for Tennis Players is a very useful book for those who wish to understand more about why the tennis ball behaves the way it does, why they should prefer one type of racquet or strings over another, or why they should play one way rather than another. The book is not exceptionally well-written, but it gets its points across. Topics covered include racquet geometry, string tensions and gauges, air resistance to the ball, how much the ball is slowed by different court surfaces, the effect of errors in racquet angle on your shots, topspin and backspin, and calculating the possiblities in playing actual "percentage tennis." Tennis Science for Tennis Players is a good book, and covers some material not in Brody's newer and larger 2004 book, The Physics and Technology of Tennis, written with Rod Cross, and Crawford Lindsey.

Racket and strings (pages 7-17):

    ...in order to change from one frame size to another while retaining similar playing characteristics from the strings, the tension divided by the string length must be kept the same. This is why the oversize racket is strung at higher tensions...
    Borg strings his [standard-size wooden] racket at 78 pounds... it is because Borg and the other top players can hit the ball so hard that they can afford to string their rackets tightly to gain other advantages; they sacrifice some of their power in doing so...
    If you do not swing your racket hard, you will often benefit from looser strings. They will tend to convert your opponents power to your power with greater efficiency. If you really want to swing out at the ball, then use a tightly strung racket. You will get that extra bit of control that hard hitters always seem to need, and the ball will still have a reasonable pace. And if you want to hit shots with a great deal of spin on them, then tight strings are an advantage.

    The dwell time of the ball on the strings should increase as the inverse of the square root of the tension; measurements made in the lab bear this out. In addition, the dwell time of the ball on the strings decreases the harder the ball is hit, because the strings become effectively stiffer the more they are forced to deform...
    The actual time of contact for a normal shot with normal string tension is about 4 or 5 thousandths of a second. By reducing the string tension and not hitting hard you might be able to increase the dwell time to about 6 or 7 thousandths of a second... These times are so much shorter than your reaction or reflex time that you cannot possibly do anything to change [the way you are hitting] the shot while the ball is on the strings...     A longer dwell time also means that the shock of the ball being hit is spread over a longer time; the magnitude of the force at any given time is therefore reduced... If you wish to alleviate arm troubles such as tennis elbow, reducing the tension of your racket strings will not only lessen the initial shock transmitted to your arm but will give you more power from the racket. You will not have to swing so hard, which is kinder to your arm...

    All one need do is reduce the tension in the strings to ensure that the ball will spend more time on the racket. Yet... loose strings decrease rather than increase a player's ability to control the ball.

    The gauge of a tennis string corresponds to the standard U.S. wire gauge thickness...
    ...since a 17-gauge string is 20 percent thinner than a 15-gauge string, the cross-section will be about 40 percent smaller. Therefore, 17-gauge strings will be almost twice as elastic as 15-gauge strings. The 17-gauge strings will not stand as much tension, will not wear as well, and will pop more often, however... if money is no object and you are willing to bring a large number of rackets whenever you play, then 17-gauge strings could be for you...

Racket stability and racket size (pages 39-53):

    When you hit the ball off-center... the racket will tend to twist in your hand and the shot will probably go awry. The property of an object, such as a tennis racket, to resist twisting is called the polar or roll moment of inertia. The moment of inertia is defined as the mass of the object times the distance of that mass from the object squared. If this moment of inertia is made larger, the racket will be less likely to twist in your hand... The moment can be increased either by increasing the mass at the edges of the frame or by making the frame wider. Increasing the width is much more effective because the added mass only increased the moment linearly, while the moment increases as the square of the width. This is why the oversize Prince racket was originally developed [by Howard Head, after he had sold control of his Head company to AMF] and is one of the reasons for its present popularity. A typical oversize racket is about 25 percent wider than a standard size racket, but it has a polar moment of inertia that is 50 percent greater... for the same off-center hit the oversize racket will twist in your hand considerably less than the standard size racket... You can also increase the polar moment by adding weights along the outside edge of the frame... as Wilson has done on its perimeter-weighted rackets...
    Although a larger [polar moment of inertia] does provide, to some degree, an inertia against twisting, it may make the racket feel more cumbersome and possibly more difficult to manuever.

    [a larger grip size will also make it easier to cope with racket twisting from off-center hits, but too large a grip can tire the hand and restrict movement on some shots]

    A second major advantage of a wider head on a tennis racket is that you will miss-hit shots by striking the frame less often...

    ...the speed of the tennis ball changes only slightly as you change the weight of the racket... This is because a 12-ounce racket is already six times as heavy as the 2-ounce ball; adding another 2 ounces to the racket does not change the ball speed appreciably... On the other hand... the speed acquired by the struck ball depends very strongly on the speed of the racket head... The extra racket head velocity that you can get with a light racket may more than compensate for its lighter weight and give you higher ball speeds.

    ...the lighter racket is more manueverable and less fatiguing to play with... [but with a heavier racket] there is less reaction back on your hand when you hit the ball...
    The lighter, head-light, low-moment racket is not only good for the big server, it also is more manueverable, which is essential for play near the net. This is the racket for the serve and volley player. The groundstroke player who plays on slow courts can use a heavier, larger-moment racket effectively...

    ...some articles and books claim that a flexible racket provides more power... there are no data to support this claim... it is stiffness, and particularly stiffness in the head, that gives a tennis racket its power [because the strings are more effective at returning ball energy in the opposite direction than the racket frame]. In addition, a stiff racket allows a player more control...
    ...so racket manufacturers try to produce less flexible frames for their top-of-the-line models.

Air resistance (pages 72-73):

    Air resistance slows the ball, and in the range of speeds encountered in tennis the force it causes is proportional to the square of the ball's speed. This means that a ball moving at 60 miles/hour would encounter four times more air resistance force than a ball moving at 30 miles/hour... Wind also creates an air resistance force... a crosswind of 20 miles/hour will exert four times as much force on the ball as a 10-mile/hour crosswind...

[For a ball hit from the baseline to land near the opposite baseline:]
initial speed
90 mph
67 mph
45 mph
speed at net
74 mph
54 mph
34 mph
speed just before bouncing near baseline
61 mph
45 mph
27 mph
total loss of speed
29 mph
22 mph
18 mph

The Magnus (curve ball) effect: topspin and backspin (pages 74-82):

    The direction of the spin of the ball is the direction that the ball will curve in the air...
    If you hit a ball with topspin, the front of the ball will be rotating downward, so the ball will curve downward (dive or sink). If you chop at the ball to give it backspin or underspin, the front of the ball will be moving up and the ball will tend to stay up or float. If the front of the ball is spinning to the right or left about a vertical axis, the ball will curve or slice to the right or left.

    No flat shot hit [at waist height] from the baseline with a initial speed of less than 30 miles/hour can clear the net, regardless of the angle at which it leaves the racket... No flat shot with an initial speed of less than 40 miles/hour can possibly go long regardless of the angle at which it leaves your racket...
    ...You cannot hit a flat (no-spin) shot at waist height from your own baseline at a speed of greater than 100 miles/hour and have it go in...

    If you give the ball topspin as it leaves your racket, you greatly increase the chances of that shot landing safely in the court. If you give the ball backspin, you greatly reduce your margin for allowable error...
    The topspin shot [also] clears the net by the most and thus gives the player the largest margin for allowable error relative to the net. The backspin shot [if it is to go in] clears the net by the least and gives the smallest margin for error.

Also see the 2004 book, The Physics and Technology of Tennis, by Howard Brody, Rod Cross, and Crawford Lindsey.

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