How slow could you throw?

A few days ago Bill Simmons posed the following question on Twitter: “Bar argument/physics question: what’s the slowest possible pitch (MPH) you could throw that would reach home plate?”

David Gassko in turn posed the question to the Hardball Times writers’ list. It was mentioned that famous astrophysicist Neil deGrasse Tyson had provided the following answer: “Slowest pitch in Baseball to reach catcher? 30mph, thrown at 45-deg angle. Any slower at any angle hits ground.”

That didn’t sound right to me, and it didn’t ring true for Sal Baxamusa, either, who noted, “30 mph at a 45 deg angle sounds to me like he’s throwing a pitch with no rotation in a vacuum. One of the other engineers/physicists can correct me if I’m wrong, but couldn’t I throw a ball slower than 30 mph if I put a good amount of back spin on it?”

I decided to investigate for myself. It turns out that Sal was correct about Mr. Tyson’s assumptions. In addition, Tyson assumed that the pitch was being released below the pitching rubber at ground level. That’s not terribly helpful. What is the answer if you make more realistic assumptions? It turns out that it’s not much different than Tyson’s result, around 27 mph, with the exact answer depending on how you fine tune your assumptions.

I used a trajectory model that incorporates the forces of gravity, drag, and spin (the Magnus force). You can download a very similar trajectory calculator from Alan Nathan’s Physics of Baseball website if you are interested in fiddling with the possible trajectories yourself.

The details of my assumptions were as follows: typical air density based on 72-degree temperature and 400-foot elevation, a drag coefficient of 0.50, a release point near that of a typical pitcher at 54.5 feet from the back point of home plate and six feet above ground level, and backspin on the pitch of 3000 rpm.

Under those conditions, a pitch thrown 26.8 mph at an upward angle of 37.4 degrees will land on the front edge of home plate. If you make the pitcher taller, the temperature warmer, the elevation higher, the backspin faster, or estimate that the drag coefficient is lower (its exact value for a spinning baseball is not well established at such low speeds), the pitch can be thrown a little slower and still make it to the plate. As such, any answer in the range 26-28 mph is probably pretty close, depending upon the exact conditions.

Bill Simmons’ question naturally led me to wonder what was the slowest pitch recorded by PITCHf/x. It turns out to be almost an impossible question to answer. Most of the slowest “pitches” in the PITCHf/x data set are errors of one sort or another, many of them from data collected in 2007. The slowest actual pitch from 2007-2010 of which I am aware is the 48-mph eephus pitch thrown by Orlando Hernandez to Luis Gonzalez on August 25, 2007. I wouldn’t be surprised if there is a legitimate slower pitch hiding in there somewhere, too.