It’s Tuesday night. The Cardinals are in that awkward phase where there is action but no stats, information but nothing of any real worth. (After all, do you really want me to break down what the Spring Training pitching groups mean for the starting rotation? No, you don’t.) My Viva el Birdos colleagues have already stolen all the good ideas for the week, a fact that I was bemoaning about to my kids.
One of them responded, “you should write about how far a home run would go on Mars!”
Then I started thinking. With my prodigious pseudo-science and quasi-math skills (see my previous work on Bob Gibson’s fastball velocity), I probably could figure out a way to translate the distance of a ball hit on Earth to Mars.
I’m a space geek. Well… let me restate that. I’m a geek. So, of course, I love space.
With Perseverance landing on Mars and incredible images coming from the surface of the Red Planet, my geek meter is currently off the charts. I watched “The Martian” this past weekend. “Hidden Figures” is queued up on my Hulu. Mars and math are on my mind.
Why shouldn’t I, in the holy name of Matt Damon’s boney bare backside, devote my evening to calculating the distance of a home run on Mars?
I should. So, let’s try to do it.
There are a lot of factors that determine how far a baseball travels on earth. Gravity makes the biggest difference. What goes up, must come down, as they say. So, when Paul Goldschmidt – despite being the horse of a man that he is – hits a baseball, it can only travel so far before it comes back down to earth.
Sure, the harder he hits it matters. More initial acceleration at the point of contact is going to lead to more velocity across the ball’s flight at a set mass (which we might or might not have with the not-so-constant density of a baseball these days).
Angle matters, too. If Goldschmidt skies a fly ball at a high velocity, it might travel a huge distance vertically – the proverbial can of corn – but it won’t go very far horizontally. The same is true of balls hit at a low angle. The velocity might be there for a time, but gravity never quits; that ball is going to drop at a constant rate. Once it contacts the earth, friction transfers that energy into pushing up daisies instead of pushing past loosely arranged atmospheric gas molecules.
If only there were a stat that calculated the optimal velocity and angle of a struck baseball to maximize distance!
MLB and Baseball Savant have a stat called “Barrels”. The smart sabermetricians over at MLB.com use real (not pseudo) science and actual (not quasi) math to calculate what exit velocities and launch angles will lead to the highest rate of production. Barrelled balls hit within a certain range of speed and rise have an extremely high chance of being a home run or an extra-base hit and an extremely low chance of being an out. They travel very far very fast.
Using the parameters set by Baseball Savant for “barrels”, we need to pick a homerun to use as our model. What we want is a ball hit by a Cardinals player that launched at around 25 degrees and was hit as hard as possible – let’s say around 110 mph. There are a few options, but this corker by Goldschmidt should make a nice choice:
For this bomb, Goldy turned on a 93-mph center-center fastball and smoked it at 111.3 mph with a 23-degree launch angle. It’s a few degrees lower than we might want, so it wasn’t the longest homer of the season. But it was one of the hardest-hit balls and it makes a nice representative sample.
That home run traveled an estimated 418 feet. On earth. With the help or hindrance of the wind that day. And the humidity. And the atmospheric conditions. And the density of the air. And the force of gravity. And the alignment of Jupiter and Uranus.
(Ok so I made that last part up just to get a Uranus reference into the article. You laughed; you know you did.)
Now, let’s take the same home run and move it from Earth to Mars. What would change?
There are some parts of this equation that we don’t want to change. For example, we’ll need to assume that the velocity of the pitch and the exit velocity of the struck ball would be constant on either planet.
That’s our first problem with this model. The Martian atmosphere is much less dense than the atmosphere on earth, so the pitcher would theoretically have less resistance in throwing the ball. His 93 mph fastball would probably be able to travel a bit faster under those conditions. Since gravity on Mars is lower, the same is probably true of Paul Goldschmidt’s swing. His bat would feel much lighter than normal. All of that could lead to more energy in the system.
Then again, neither the pitcher nor Goldschmidt could survive on Mars without a restrictive space suit. So, we’re making all kinds of unrealistic assumptions here. We’re going to ignore the impact of the spacesuit and we’re going to set the current pitch and exit velocity as constants.
Quasi-science finds a way!
What about launch angle? I don’t know why that would change regardless of what planet the ball was hit on. Vectors are vectors. The same pitch hit in the same way at the same velocity will likely have the same angle.
With quasi-scientific assumptions about exit velocity and launch angle in place, all we need to know now is how the difference in the force of gravity on Mars compared to Earth will affect the ball flight.
NASA comes to the rescue. The force of gravity on a planet has a lot to do with the density of that planet. The earth has a higher density than Mars. So it has a higher gravitational force than our nearest neighbor. What is that difference? NASA tells us that the “surface gravity” of Mars is 3.71 m/s^2 compared to 9.8 m/s^2 on earth.
It seems like if we know the launch angle of a struck baseball and its exit velocity and the gravitational coefficient of the planet the baseball was hit on, we should be able to calculate how far the ball would travel. Sure, it might require some kind of crazy calculus and trigonometry but it feels doable.
A few years ago Purdue Ph.D. student and golf-enthusiast Jocelyn Dunn set out to run this same experiment with golf drives. Being the actual scientist that she is, she even went up onto the mountains of Hawaii to hit drives with a spacesuit on to see how much the suit restricted her movement and ball distance. You can see her experiments here.
Dunn also developed the mathematical formula that we can use to translate her experiment from golf to baseball. Here it is:
I prefer quasi-math to real math, and this had the putrid stink of real math all over it. So, I turned to my baseball analytics Twitter DM group with a cry for “help!”
Zach Gifford’s response? “Woof.”
John LaRue ignored me.
Ben Cerutti — a math teacher — passed.
I should have DM’ed Ben Godar… he wouldn’t do the math but he would have given me some really tight butt puns.
Matt Graves over at Redbird Rants came through, breaking out his big brain and a TI-84 scientific calculator.
First, we had to do some conversions. The exit velocity of Goldschmidt’s homer was in miles per hour. The gravity constant of Mars is meters per second squared. 111.3 mph becomes 49.76 m/s. That has to be squared so our times – seconds – match throughout the equation.
“Sin” and “cos” are trigonometric functions that are used to calculate distances by angle. At one point in the distant past, I knew what those things meant. Now, Matt and his TI-84 remember what I’ve forgotten, so I’ll just have to trust them. The 23-degree launch angle goes into the equation. They are sined and cosined, and all of that is multiplied together and by two (instead of multiplying the bottom of the equation by ½).
Then we divide it all by the gravitational coefficient of Mars. The seconds squared cancel out. The “m” on the bottom erases one of the “m^2’s” on top, leaving us with just one m = meters, a measurement of distance.
That’s our answer: the distance the ball would have traveled on Mars is 480.09 meters.
Meters are for cricket-playing socialists, though. So back to Google. If we convert meters into good ‘ol American feet, we have our final answer.
Paul Goldschmidt’s 418-foot Busch stadium homerun on Earth would have traveled a whopping 1575 feet on Mars. (Give or take!)
If that seems like a long way, well, it is. Let’s put it back into context. Goldschmidt’s home run landed in the corner of the left-field bullpen at Busch stadium. If we ignore vertical obstacles – like the stands, Ballpark village, and flocks of birds – and bring the Martian distance back to earth, then the ball would have landed on the steps to the Old Courthouse on Market Street.
That’s one interstellar blast to left!
If you are even more of a geek than me, you can take that same equation and run the numbers for any other home run you want. The Cardinals’ longest homer of the 2020 season was hit by Brad Miller at 446 feet, with a 107.4 mph exit velocity and a 26-degree launch angle. I suspect it would produce an even better result. You can tweak the search parameters on Baseball Savant to any launch angle and exit velocity range you want to find the ideal home run to move to Mars for maximum distance.
If you have any ideas on how to improve this approach or ways to tweak it, include those in chat! I’m no math or physics expert, so take this and run with it and let me know your results.
UPDATE: from bobohilario in the comments. How awesome is this:
Here is a simulation I made in Mathematica that includes drag and lets you manipulate launch angle
The value for the drag coefficient is a complete guess, but the effect is very small. At the same 23 degree launch angle I get a similar result.
However, in a low drag environment the ideal launch angle is close to 45 degrees. With that angle and Goldy’s 110 mph the ball went over 2000 feet.
Converting from Martian feet to Earth feet that is over 2000 feet!
Thanks to all who contributed to this article, especially Jocelyn Dunn for the formula, Matt Graves for the math, the Statcast DM group for the laughs, and Baseball Savant for the data and video.