29 April 2013

The Path of a Home Run

As of Sunday, 28 April 2013, the longest home run hit in the Major Leagues so far was hit by Anthony Rizzo of the Chicago Cubs.  The left-hander hit a shot off righty Alexi Ogando of the Texas Rangers on a rainy day in Wrigley Field on the 18th of April in the bottom of the third inning with nobody out and a man on first.  Video of the home run may be found here.  Though the ball hit the back of the bleachers, an estimate of where the ball would have landed on the ground (really North Sheffield Avenue) has been made.  A great website that tracks all Major League home runs is Hit Tracker, which may be found here.  If one sorts all home runs by true distance, one finds that Rizzo's shot would have traveled 475 ft (145 m) horizontally.  Also provided are the launch speed off the bat, which was 115.0 mph (51.4 m/s = 185 km/hr), and the launch angle measured from the horizontal, which was 24.5 degrees.  The maximum height of 86 ft (26 m) is also given.

With all the great data provided, anyone can play with real home-run trajectories.  I describe how one may do this in an aerodynamics review article I wrote that just appeared online (click here to access the paper).  By choosing the appropriate drag and lift coefficients, I can fit a model trajectory to one that matches the data on the website.  I use constant aerodynamic coefficients as a first approximation, and I ignore the tiny difference between initial and final heights (about a meter, which is only about 0.7% the size of the horizontal range).  My model doesn't include wind or effects of rain, but it does give quite reasonable estimates of the sizes of the forces on the ball while in flight.

The buoyant force on the baseball was just under 0.2% of the ball's weight (my online article has a typo; the buoyant force I consider there should have been 0.15% of the ball's weight instead of 1.5%).  Clearly that force isn't a big player here!  The drag force is about 1.5 times the ball's weight just after the ball left Rizzo's bat.  If you solve a projectile motion problem with "ignore air resistance" in the problem statement, know you are not solving a realistic problem!  The initial lift force, which is due to the roughly 2000 rpm backspin the ball had when it left the bat (only around 500 rpm when the ball landed), was about 80% of the ball's weight.  Ignoring the effect of the ball's spin is also highly unrealistic!

The graph below shows three trajectories (click on the image for a larger size).
The dotted curve is what the trajectory would have looked like in vacuum.  That trajectory is nearly 41% too far.  If one includes drag, but not lift, one gets the dashed trajectory.  But that one is just over 17% too short.  The Goldilocks trajectory is the solid trajectory, which is my model of Rizzo's home run.  I missed the actual range by just 0.005% and the actual maximum height by 0.05%.  I could, of course, tweak my drag and lift coefficients to do even better, and I could include the slight difference in launch and landing heights, but pursuing this much further is silly because I don't know the true atmospheric conditions on that rainy day in Chicago.  The fun part is getting a trajectory that matches the real-world trajectory quite well and then studying the forces involved.

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