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.

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Comments

  1. hansioux said...

    I know it says bar physics, but there should be a lower bound by physics of how slow you can pitch a ball and reach home plate.

    If a pitcher throws a ball straight forward without any upward or downward movement, it should take the same amount of time to touch the ground if he just drop the ball from the height of his release point.

    That amount of time is the longest a ball can take to reach the plate.  so 60’6” divided by that time is the slowest possible pitch to reach home plate just as it touches the ground.

  2. Mike Fast said...

    Hansioux, it’s true that if you restrict yourself to giving the ball no upward or downward velocity, and like Dr. Tyson, you’re pitching in a vacuum, the time to hit the ground will be the same no matter the horizontal velocity. 

    However, the question did not say that you could not give the ball any upward velocity.  That makes a great deal of difference for the ball’s flight time.  If you throw the ball up, it obviously takes longer to hit the ground than if you simply drop it.

  3. J_ball said...

    It will be interesting to see a guy who use a 26-30 mph “fast”ball as a staple pitch . along with a 70 mph curve ball… to fan MLB hitters…

  4. Mike Fast said...

    Ike Hall points out in the discussion at the Book blog that my estimate for backspin seems too high:
    http://www.insidethebook.com/ee/index.php/site/comments/you_need_to_throw_27_mph_to_reach_home_plate/

    My response to him there:

    Ike, you may be right to some extent.  There is some portion of the angular velocity which is provided by the velocity of the arm.  The rest is provided by the snap of the fingers.

    My thought was that you could get quite a bit from the finger snap/roll if you were specifically trying to emphasize that.  A major league pitcher can get about 1000 rpm more from the finger snap on a fastball relative to the changeup which is seated deep into his palm, and he’s doing that while trying to make the two pitches look as much the same as he can.  You do, in fact, see that in the PITCHf/x data.

    But then we leave the realm of the PITCHf/x data to the realm of my guesstimates.  I guessed that a person really trying to emphasize backspin above all else could maybe could get 1500-2000 rpm from the finger snap/roll.  That’s equivalent to a 13-17 mph difference between fastball and changeup due to grip/release, which doesn’t seem outlandish for someone trying to emphasize that with no regard for deception.

    I further guessed that the translational arm motion would provide another 1500-2000 rpm or so.  That’s the portion which will be linearly proportional to the velocity of the ball.  Looking at it now, that estimate is probably too high by a factor of 3.

    So an estimate of backspin in the 2000-2500 rpm range is probably more realistic.  That would bump the minimum release speed up to 27.4 mph.

  5. Neil deGrasse Tyson said...

    Happy to see the slow-pitch subject continued here.  My tweeted calculation was indeed for a projectile moving in a vacuum, released from ground level, travelling 60ft 6in.  But I also considered more realistic conditions before I posted the nice round 30mph figure.

    Air resistance would require a higher speed.  The fact the ball gets released at an elevation higher than the catcher’s mitt would allow a slower speed.  A backspin to provide more lift would allow a slower speed.  The ball’s release point is not at the pitching rubber but at least 3-feet down-range from it, also allowing a slower speed.  And the catcher’s mitt is not at home plate, but several feet behind it, requiring a faster pitch.

    So I concluded these details are not reproducible from pitcher to pitcher, catcher to catcher, or even from to pitch to pitch.  And they don’t make much difference in any case, as already noted in this thread.  So offering a slowest-speed with any more precision than a simple “30mph” would be scientifically misleading and in fact disingenuous.  -NDTyson

    ( Tweeting at:  http://www.twitter.com/neiltyson )

    p.s.  I was mildly chastised for Tweeting baseball during the World Cup.  So keep all this among ourselves.

  6. Dave Studeman said...

    I’m not sure, Mike, but I think you’ve just been dissed by a physicist who’s been on the Daily Show.  Cool!

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