Bob Larson -- "Physics Footnotes Pt. 2" [Archive] - MensTennisForums.com

Bob Larson -- "Physics Footnotes Pt. 2"

tennischick
11-07-2004, 02:42 PM
Physics Footnotes, Part II

Our first Physics Footnote (way back on July 16, 2003) was mostly about air resistance. But, believe it or not, there is quite a bit more we can say about this topic.

Air resistance shows up in all sorts of places in tennis. It explains part of why players try so hard to put spin on a ball -- a ball without spin isn't stable and will tend to flutter around. If you played tennis in a vacuum, on a hard enough court, you could just whack at the ball. But not on earth's surface; you want to brush through it.

Air resistance also has the interesting side effect of producing unnaturally high radar gun readings. If tennis were played in vacuum, players would have to hit more slowly, because the air wouldn't hold their balls in the court. So you can thank earth's atmosphere for more than just giving you something to breathe -- it also makes Ivo Karlovic an amazing figure rather than just another guy with a less-than-perfect ground game.

And speaking of radar guns, we talked about them once, mentioning how air resistance affected their readings. We also mentioned, as an aside, that they measured wide serves as being slower than serves down the T.

Now there are two reasons why this can be true. It must be remembered that every "power" shot in tennis is, in a way, trying to thread the eye of a needle: It has to be high enough to go over the net but not so high as to land beyond the line (service or baseline). And since the wide serve has to go over a higher part of the net, it might seem as if it has to be hit slower.

Not really. Think about a tennis court, and the paths followed by the two serves:

+-------------+
| |
| |
+------+------+
| | |
| | |
| | |
+------+------+
| | |
| | |
| | |
+------+------+
| |
| |
+-------------+

Imagine drawing a line from the service line to the far T. On this graph, that's eleven "units" long. Now draw a line from the corner of the service box. (In fact, we've drawn both below, with T representing the line down the T and W the wide serve.)

+-------------+
| |
| |
+------+------+
W T |
W T |
|W T |
+-W----T------+
| W T |
| W T |
| W T |
+---W--T------+
| W T |
| WT |
+------T------+

It's probably fairly clear that the line W is longer than the line T. Not vastly so (a little work with the Pythagorean Theorem shows that it's only 61.5 feet, compared to the 60 feet from service line to T) -- but that's enough to gain back most of the distance lost to the higher net. Indeed, by standing a little wide of the center, the server can gain it all back.

The real reason wide serves are slower lies in the way the radar gun works. It measures only motion directly toward the gun. If theta is the angle between the ball and the line straight up the T, and the initial velocity is V, then the velocity as measured by the radar gun will be V*cos(theta). The maximum theta for a ball hit from the exact center of the court to a sideline is about 12.5 degrees. In practice, it can be closer to 15 degrees -- which means that cos(theta) can be as small as .96 (for a ball straight down the T, cos(theta) is 1.0). That multiplier, close to 1.0 though it is, translates to about 5 miles per hour off a 140 mile per hour serve. Not huge -- but it will result in serves down the T being a bit bigger.

And since we're talking about the serve, let's bring out a second point. Most of these things come up, we should note, as we watch matches and think, "Why?" Why, for instance, do players lunge for serves that go into the net, or way long? If their eyes are as good as they are, shouldn't they be able to read serves?

The answer, really, is no. There is a key point here: Whether a serve is in or not depends on where it is moving and how fast it is moving. If a ball is aimed in a particular direction and it moving too slowly, it will end up in the net; gravity sucks it down. If it's pointed the same direction and moving too fast, it will go long; gravity doesn't have time to pull it into the court. Only if the speed is "just right" (give or take a little for margin of error) will it end up in the service box. So to judge whether a serve is going in, you have to judge how fast it is moving.

But this is much harder than it sounds. Human beings aren't radar guns. A person measures speed in one of two different ways: By parallax, or by size comparison. Parallax is a form of triangulation: Each eye looks at the object, and the brain examines the image seen by each eye, and in effect calculates the angle to the object. By comparing the different angles seen by the two eyes, the brain can calculate the object's distance. Parallax is very reliable when it works at all; assuming you can measure the angles, you can't get the result wrong. (Parallax, in fact, was used to measure the distance between the earth and the sun, and more recently used to measure the distance to the nearest stars.) But there is a problem: Your eyes are very close together. A tennis court is long enough that you can't measure distances by parallax. That pushes you back on the other method: Size comparison. How big does the ball look compared to all the other stuff around it? Or, since you know how big a tennis ball is, how big does it look compared to how big it would be if it were, say, 20 feet away? This isn't an easy method of measurement. You have to (unconsciously, of course) measure the change in the apparent size of the ball. Given time, you can measure things that way. But serves these days are so fast that time is precisely what you don't have. So players lunge at serves, because if they wait until they are sure they should or shouldn't, it's too late.

And since we're talking about serves, let's conclude with one more discussion on this topic. A few years ago, this was a real fear: That most aspects of the game would decay because the serve would become so dominant. Power would be everything.

So what happened?

The answer is, of course, only a guess -- but it is a very "physical" guess. What happened is, the serve in fact saved men's tennis.

It's worth thinking about the state of the women's game. Modern equipment -- titanium racquets and hardcourts -- have led to a situation where every player's best shot is her return of serve, and every player feels pressure on her serve. If a match ends in a 6-0 6-1 or 6-1 6-1 blowout, it's likely that the loser's only games came on breaks.

So why hasn't this happened with the men?

The likely answer is, the serve.

The crucial point is limits: All shots are limited by court geometry. But the speed of a groundstroke is absolutely fixed (you can be Godzilla and you still can't hit a 500 kph groundstroke; such a ball is guaranteed either to go into the net or go long), whereas the speed of a serve is constrained primarily by the height at which the ball is hit. The taller you are, the harder you can serve. And since men are, on average, taller than women, they can serve better. That gives the server that very slight extra advantage over the returner. It's enough to keep the men relatively honest. Even so, on hardcourts it is the baseline bashers (the Agassis and the Hewitts) who have tended to dominate. Even Andy Roddick fits this mold, really; he's a basher with a big serve. But the bigger serves that most men can develop -- because they're taller -- can keep the Agassis and Hewitts honest. Especially on surfaces like carpet and grass.