∫ Things I learned from teaching undergraduates: Tennis Ball Topspin Edition

The Youtube clip above shows Roger Federer hitting a topspin forehand in slow motion. The ball is moving too fast after he hits it for you to be able to see the direction of its spin, but you can tell pretty clearly from the motion of his racket that he's hitting a little bit beneath the ball and then sliding the racket upwards along the backside of the ball at the same time as he follows through. The result is the the ball will be spinning with what tennis players call "Topspin."  It's a rotation where the ball the top of the ball is moving in the same direction as the ball is flying, and the bottom of the ball is moving the opposite way.

Topspin is a huge deal. Part of the reason Roger Federer is so good is that he hits with 2,500 RPM of topspin, which is more than either Pete Sampras or Andre Agassi (only Rafael Nadal hits with more, although it's been argued that he actually hits with too much!)

Now, like anyone who's ever taken a tennis lesson, I know tennis reasons the pros hit most of their shots with topspin. Coach after coach after coach has told me over the years that there are basically two principal reasons: first, a ball with topspin will "kick" upwards when it bounces (I think it's pretty easy to see why) and that can make it harder for your opponent to hit, because they wind up having to hit it up around their ears. But then there's this second reason: Topspin is supposed to make the ball "drop" sooner, meaning you can hit it higher over the net and still get it to land inside the court. Being able to hit higher  in turn means you have less chance of hitting it into the net, and it means that you can hit it a little harder. Seems like a good deal all around.


But one thing I definitely did not know until a couple of days ago is why topspin causes the ball to move like this. After all, why would spinning make a ball want to fall downward faster? It's not like it makes it any heavier.




Well I finally learned the yesterday when I happened to run into Dr. Chang Kee Jung, who teaches a course called "The Physics of Sports" here at Stony Brook (see this post). I mentioned to him that I had always wondered about what sort of magical fluid dynamics cause this to happen, and he was kind enough to explain it to me, so I'm going to try to pass along the favor.


The key is something called the Magnus effect, which is not only responsible for why topspin works the way it does, but also for why curveballs curve, why golf balls can "hook," and a whole host of other phenomena. Basically, every time you have a ball spinning as it travels through air, it's going to want to move a little bit in one direction or another. But why?


There's actually two components to the answer. The first one is pretty easy to see, but, unfortunately, is the smaller of the effects. When a spinning ball moves through air, some of the air that flows past right along the ball's surface gets dragged along with the ball's rotation. In the case of a tennis ball, it's grabbed by the "fuzz" on the ball's surface. This "dragged" air gets separated from all the other air flowing past, creating a little whirlpool of air around the ball, as in the picture below.

The Magnus effect, part 1

Now, as you can imagine, the air on the bottom of the whirlpool is moving faster than the air at the top, because it's being pushed along by the air behind it. The air at the top has to fight against the oncoming wind. But Bernoulli's principle tells us that faster moving air has lower pressure, meaning there's more pressure above the ball than below it, which will naturally force the ball downward. In this sense, the spinning ball is acting a little bit like an upside down airplane wing.


Unfortunately, while this sounds great in theory, experiments have shown that the amount of air in this whirlpool is pretty small, so the pressure from the Bernoulli effect is small, too small to explain the drop of a topspin lob. The rest of the effect comes from a more complicated process: one the boundary layer whirlpool is established, it will tend to "grab" at the rest of the oncoming air, meaning that as air flows past the ball on the bottom, it will linger slightly longer, pulled along by the whirlpool, and then finally be released in a little bit more of an upward direction. Conversely, on the top, the whirlpool pushes against the oncoming air just a little bit, meaning it gets forced away from the ball slightly sooner than otherwise, and again in a bit of an upward direction:

The Magnus effect, part 2

But now, Mr. Newton has something to say to us; namely, "Actioni contrariam semper et æqualem esse reactionem," or if you prefer, "To every action there is always an equal and opposite reaction."If you're floating in a swimming pool and you push some water to the right with your hands, you'll move a little to the left. If you're skating at an ice rink and you toss your coat into the bleachers, you'll slide the other way. And so, too, since the ball is tossing some of the air upwards, it's going to have to move a little bit downwards. And voila! That lob you all thought was going to land three feet long dropped just inside the baseline, courtesy of topspin.