The Physics of Identical Homers

It might seem reasonable to assume two balls hit at the same exit velocity and launch angle should travel the same distance. For example, on April 12 of this year in the eighth inning in Miami, Andrew McCutchen torched this 3-2 pitch for a round-tripper:

Meanwhile, in Los Angeles, Yasmani Grandal sent a 1-0 change-up into the Pavilion in the fifth:

According to Statcast, both balls left the bat at 101.3 mph with a launch angle of 29 degrees. Both dingers traveled 401 feet before returning to Earth. These could be called “identical homers.” Let’s define balls hit with identical exit velocities and launch angles to be “identical launches.” All identical homers are also identical launches, but not all identical launches become identical homers.

With this in mind, let’s examine the homers recorded by Statcast in 2019 through the All-Star break. There were 3,691 of them. That means there were nearly seven million possible pairs of home runs. Of that seven million, there were only about 3,300 identical launches and just 43 identical homers. The McCutchen/Grandal pair were the only pair of identical homers hit on the same day–that’s why I picked them as an example. Identical homers are a very rare coincidence; on the same day, even more so. The rarity of identical homers can be understood by examining the physics of the flight of the ball.

Below is a table of a particularly interesting set of identically-launched homers, all of which left the bat with an exit velocity of 107.0 mph and a launch angle of 23 degrees. The table includes the date, batter (including handedness), ballpark, distance traveled, the landing direction, and a link to a video of the home run.

Dwight Smith Jr.’s bomb was his first career grand slam. The same is true of Ryan O’Hearn’s blast, which was one of two grand salamis for Kansas City that day. Keston Hiura’s round-tripper was his first in the big leagues. Tommy Pham’s was his first homer of the year; he hit his second later in the day. And the Grand Old Man–Curtis Granderson–crushed a pinch-hit job in the top of the ninth to tie the game.

From a baseball perspective, these truly are a notable collection of long balls, all with the same exit velo and launch angle. They are also impressive from a physics perspective because the home run distance varies dramatically from 366 feet to 442 feet. That’s a 76-foot range, vividly illustrating that identical launches do not automatically result in identical homers.

To investigate the source of these distance variations, let’s compare the longest (Smith’s) to the shortest (Hiura’s). The usual culprits are weather and ballpark elevation (think Denver). These quantities change the density of the air through which the ball must travel. The higher the density, the thicker the air–and the shorter the flight.

The table below lists the temperature and wind for these two dongs from the box score, the barometric pressure and humidity from somewhere near the park (WeatherUnderground), and the ballpark’s elevation.

Using the data above and Alan Nathan’s Trajectory Calculator, one can estimate the actual flight path of the ball. The first step is estimating the spray angle off the bat using the rumored equation and confirming it (more or less) with the video. Now, the backspin and sidespin can be adjusted to get the correct distance. There is a bit of subjectivity in this process because both the backspin and sidespin affect the distance.

Less judgement would be needed if Statcast provided the top and side spin of the batted ball or the actual landing location of the ball. Statcast gives the distance where the ball would hit the ground if it were unobstructed, but there is no public data on the other position coordinates (the x-y position, if you like).

The temperature can’t be the cause of the difference in the homers’ distance–it just happened to be the same for both long balls. The ballpark elevation can’t really be responsible either, since Hiura’s shorter home run was actually hit at a higher elevation than Smith’s ball. If both balls were launched at a 33-foot elevation, Hiura’s homer would have traveled four feet shorter–not longer–than it did.

Since humid air is less dense than dry air, increasing Hiura’s humidity to match Smith’s would again result in a shorter distance, not a larger one. If you care, it would only increase the distance by a paltry two-tenths of a foot anyway.

The barometric pressure felt by Hiura’s ball was less than that of Smith’s, so the air density was even smaller for the shorter homer. Once again, this is the wrong direction to explain the difference. Hiura’s ball would have gone about three and a half feet less in the higher pressure of Camden Yards that day.

We seem to be heading further from an explanation, not focusing in on one. Perhaps the wind will provide some answers. If both balls were hit without wind, Hiura’s would have gone 339 feet while Smith’s would have traveled 436 feet. Now, we’re almost 100 feet apart…for crying out loud!

It might be time to take stock. It must be true that two identical balls hit at the same elevation in the same weather at the same launch angle with the same exit velocity will travel the same distance unless physics is broken. So, what the heck is going on?

The answer? The spin on the ball. Smith smacked the ball just to the right of center, while Hiura flared his ball down the left-field line. Every good outfielder knows balls hit down the line tend to veer off toward foul territory while balls hit toward center don’t fade nearly as much.

This is due to the fact that the ball-bat collision imparts almost no sidespin to the ball when the bat is level and perpendicular to the trajectory of the incoming pitch–a ball that will head off toward center. However, when a ball is hit down the line so the bat is not perpendicular to the pitch direction, the result is sidespin on the ball. In golf vernacular, one might say Hiura hit a “hook” while Smith drove it “right down the fairway.”

Another factor is that Smith hit a high strike, so his bat was almost certainly a bit below the center of the ball, creating backspin. This backspin causes a ball to stay in the air longer than it would otherwise, giving it more time to travel further. Meanwhile, Hiura’s hit a low strike and likely put topspin on the ball. The topspin causes the ball to stay in the air for a shorter period of time and reduces the distance.

It is pretty clear in the video of Hiura’s ball that it is swerving toward the foul pole and dropping like a rock as it lands in the bullpen. This is quite different than Smith’s bomb that settled down into the stands in the usual way.

So, identically hit balls must not only have the same exit velocity and launch angle, but also the same spin to be truly identical launches. In addition, if they have the same weather and elevation, they will travel the same distance.

Physics is saved! Unless, of course, somehow the balls are different, but that is another developing story.

Note: Thanks to the Effectively Wild podcast and a question from a listener for the inspiration for this article.