The physics of foul tips, from Alan Nathan

A few days ago, I asked on Twitter whether a foul tip could possibly go faster than the incoming pitch. The question was prompted by the news that the Marlins had used a pitching machine to fire 100 mph fastballs at their home-plate fish tank to ensure that it could take the punishment. I wondered whether a pitch that came in at, say, 99 could somehow be sped up a few miles per hour and get past the threshold the Marlins had tested, murdering a thousand poor fishies in a single terrible incident. After a lot of guess-work on the part of a variety of people who know varying amounts of physics (with most guessing no, and at least one random Google result saying yes), someone suggested that I ask Alan Nathan.

Well, duh. I should have done that in the first place. Dr. Nathan is, of course, the world’s preeminent physicist of baseball. He graciously responded in the email that follows (from France!) and even more graciously gave me permission to post it. Here it is:

The primary physics involved is the frictional force as the ball skids along the surface of the bat when the bat just skims the ball, leading to a foul tip directly away from the pitcher.  If the frictional force is away from the pitcher, the ball will speed up. And if toward the pitcher, it will slow down.  The latter is the more likely result, as we will see from the example below.

Suppose the ball is moving horizontally at 85 mph and the bat is moving horizontally in the opposite direction at 70 mph (both realistic numbers).  So, the *relative* ball-bat velocity is 155 mph, pointing away from the pitcher.  Suppose also the ball is spinning with topspin, so that the bottom of the ball is rotating toward the pitcher.  And finally, suppose the bat just nicks the bottom of the ball, leading to a foul tip.  The surface of the ball slides along the surface of the bat at a speed of 155 mph minus the contribution due to the topsin.  For typical amounts of topspin (say, 2000 rpm), the contribution of the spin is really very small (17 mph for 2000 rpm), so that the motion of the sliding is away from the pitcher.  Friction must therefore act in the opposite direction, which slows the ball down.  The spin rate would have to be about 18000 rpm for the sliding to be toward the pitcher, in which case friction would act away from the pitcher, resulting in the ball speeding up.

In fact, such a thing is more likely to occur with a ground ball hit with a lot of topspin.  When the ball hits the ground, it can actually speed up.  Still, it requires a lot of topspin and is not likely to happen.

There you have it!