The Magnus Effect Equation

On the left side of the Magnus effect equation is Fm. This is the total downward force that acts on the ball, or the total effect of topspin placed on the tennis ball. The larger this number, the better. On the right side of the Magnus force equation, there are a bunch of variables that are multiplied together, so we can see that the total downward force depends on a number of different things. Let’s look at each one individually.

First, we have 1/2p. p stands for the density of the air that the tennis ball is moving through. Given that we divide the density of air in half here, and that air is not a dense substance to begin with, it would logically seem that air density really isn’t a huge factor in determining the total topspin effect we can get out of a tennis ball. Air density might vary because of temperature (the hotter it is outside, the less dense the air is) or altitude (the higher above sea level you go, the less dense the air gets). Indeed, various experiments have shown that as far as a spinning tennis ball is concerned, even relatively large natural variations in air density don’t really affect how hard the force generated by topspin can push a tennis ball downwards.

The next coefficient in the Magnus force equation is V2. V stands for the velocity at which the tennis ball is moving through the air. How hard you hit the ball is important in determining the downward force from topspin because the faster the tennis ball goes through the air, the greater the forces of drag acting on it, and thus the greater the resulting downward force from topspin. This explains how players are able to hit serves at 155mph and still place the ball inside the service box, or how Gael Monfils can hit a forehand at almost 120mph and still have the ball land in. There is a limit to this effect due to the particular characteristics of the tennis ball (fuzz and seam lines). Past a certain speed, the velocity of the ball becomes relatively less important, not to mention the fact that hitting harder is more difficult to control. Also, if you hit a serve at 155mph, the ball spends 25% less time in the air than a ball hit at 100 mph, which means the ball has less time to “move” downward due to any spin you place on it. To get the maximum effect from a topspin serve you only need to hit the ball at about 100 mph, well within the abilities of skilled club-level players (3.5-4.0+). The same holds true in baseball: Because of the particular characteristics of the baseball (stitches, surface texture) you actually get more ball deflection force from spin by throwing a curveball at 90 mph than you do at 100mph.

A (also sometimes abbreviated “S”) stands for the cross-sectional area of the tennis ball, which is a constant number because all tennis balls are the same size.

Cl stands for the coefficient of lift, which is a number that is derived from another equation. We won’t go into the details of that equation here because it involves differential calculus and a fluid dynamics coefficient called the Reynolds number. Suffice it to say that the coefficient of lift is derived largely from the angular velocity of the tennis ball, i.e. how fast it is spinning. Hands down, the amount of topspin placed on the ball is the single most important factor in determining how hard the ball will be pushed downward.