Projectile motion puzzle

This problem is inspired by Jackie Bradley, Jr., outfielder for the Boston Red Sox, who last week in warm-up threw a baseball from near home plate over the 17-feet-high wall in deep center field, 420 feet away.  (Here is a video clip of the throw.)

It’s a pretty amazing throw… but just how amazing is it?  That is, how hard would you have to throw a baseball to clear a 17-foot wall 420 feet away?

This is an interesting question in its own right, with the usual appeal of encouraging both pen-and-paper as well as computer simulation for a solution.  I’ll get to the answer shortly– but while working on it, I encountered an interesting relationship between some of the variables that has a nice geometric interpretation, which I think is best illustrated with the following slightly different version of the problem:

Problem: Suppose that you are standing on the outer bank of a moat surrounding a castle, and you wish to secretly deliver a message, attached to a large rock, to your spy inside the castle.  A wall h=11 meters high surrounds the castle, which is in turn surrounded by the moat which is d=19 meters wide.  At what angle should you throw the rock in order to have the best chance of clearing the wall?

At what angle should you throw an object to clear a wall 19 meters away and 11 meters high?

At what angle should you throw an object to clear a wall 19 meters away and 11 meters high?

The intent of the large rock is to allow us to ignore the relatively negligible effects of air resistance, thus preventing the calculus problem from becoming a differential equations problem.

We can’t afford to do that with a baseball, though.  Coming back to the original problem at Fenway Park, there are two important atmospheric effects to consider.  First, air resistance significantly increases the speed at which Bradley must have thrown the ball to clear the outfield wall.  But second, the Magnus force resulting from backspin on the ball (also responsible for curve balls and surprisingly hard-to-catch pop-ups) actually makes the ball travel farther, thus decreasing the required speed compared with a ball thrown with no backspin.

Accounting for both of these effects, by my calculations (which I can share if there is interest), Bradley would have had to throw the ball at over 105 miles per hour, at an angle of a little over 30 degrees.


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