The Physics of Identical Homers

Keston Hiura’s identical home run traveled the shortest distance of any home run with the same launch angle and exit velocity. (via Ian D’Andrea)

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.

Some Identical Homers in 2019
Date Batter Bats Ballpark Distance (ft) Direction Video
6/26/19 Lourdes Gurriel Jr. R NYY 415 LF link
5/31/19 Dwight Smith Jr. L BAL 442 R-CF link
5/19/19 Keston Hiura R ATL 366 LF-line link
5/7/19 Ryan O’Hearn L HOU 373 RF link
4/10/19 Tommy Pham R CWS 409 CF link
4/7/19 Curtis Granderson L ATL 439 CF link

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.

Environmental Factors for Two Identical Homers
Batter Ballpark Temp Wind (mph) and Direction P (in) Humidity Elevation (ft)
Smith Jr. BAL 83˚ 2 mph out to RF 29.8 54% 33
Hiura ATL 83˚ 10 mph R to L 28.9 42% 1001

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.

David Kagan is a physics professor at CSU Chico, and the self-proclaimed "Einstein of the National Pastime." Visit his website, Major League Physics, and follow him on Twitter @DrBaseballPhD.
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3 years ago

Very cool article!

3 years ago

The next question is whether there are public models for estimating spin, since that seems to be a huge lever for flyball distance. And do some players do a better job at creating optimal spin on their flyballs that would affect the expected HR/FB rate?

Alan Nathanmember
3 years ago
Reply to  Werthless
A N0n
3 years ago
Reply to  Werthless

My guess would be that it’s mostly a function of what pitches a hitter swings and makes contact with? So a player like Jose Altuve who likes the ball up in the zone would, on average have more backspin on their flyballs vs a lowball hitter like Anthony Rizzo.

Edit: There are some players I have noticed who seem to create backspin consistently on their flyballs. Justin Turner comes to mind.

Alan Nathanmember
3 years ago

Good article, David. When you ran the Trajectory Calculator, what backspin and sidespin did you find accounted for the distance? Do they agree with your backdpin/topspin explanation for Smith/Hiura?

3 years ago

Why are there ads in front of every highlight that are equally long as the highlight themselves? Possible to re-embed as a gif?

3 years ago

I didn’t realize that spin affected hit balls this much in total defense. Old-school hitting coaches always want their guys to slice down in order to create backspin, in their words. That said, the new-school hitting approach— i.e. uppercutting slightly— should also generate backspin by making contact slightly under the ball.

I wonder if the slicker surface & lower seams of the new ball also assists w this— while some pitchers have difficulty imparting as much spin as before, how much spin hitters impart is less affected, but the change also means that the rate of spin loss is lower and so fly balls (particularly long, high ones) ‘carry’ more.

A N0n
3 years ago

Great article and thanks to Alan Nathan for his continued research on the subject. Baseball savant’s distance readings seem very suspect at times. It definitely doesn’t seem to be accurate when there is wind.

An example:

Projected distance is 381 yet the ball would have hit the base of the wall which is listed at 400 feet?

A N0n
3 years ago

Also, what kind of spin causes flyballs and high launch angle liners to “slice”?

3 years ago

Baseball is wondrous. Physics is wondrous. I wish I was good at just one of them.

3 years ago

Very interesting.

Sorry if this is a dumb question, but where do you get the data for x0(ft), y0(ft), z0(ft) and wg(rpm) in the Trajectory Calculator. Presumably Statcast is the source, but which fields do those four Trajectory Calculator fields map to in Statcast?

3 years ago

“All identical homers are also identical launches, but not all identical launches become identical homers.”

Surely identical homers do not have to be identical launches. The same factors that turn identical launches into non-identical homers could work the other way as well. In fact, we wouldn’t have to appeal to those other factors. There have to be different combinations of EV and LA that result in the same distance. There have been nearly 6000 HR hit this season. The longest one was <200 feet longer than the shortest, so there must be literally dozens of HR that travelled the same distance, within a foot. Obviously most of them resulted from a different combination of EV and LA.