Are the Baseballs Okay?
Playing around with the physics of manipulating games
Between posting about The Pip’s baseball exploits and following a bunch of New York Yankees fan accounts, my ex-Twitter feed is now chock full of baseball content. (Which is mostly good, because the alternative is politics, which is stupid and annoying…) Last night, there was one tweet in particular that caught my eye:
The text here, if you can’t get it from the image and don’t click links, reads:
Since 2015, there have been 195 batted balls struck between 107.0 and 107.4 MPH with a launch angle between 29 and 30°, including this one. They've traveled, on average, 421.4 feet.
This was only the 5th such ball to go 380 feet or fewer, and the 14th to not be a home run.
If you’re not a sports fan, the context for this is that the twitterer in question, Michael Kasper, is engaging in one of the great annual traditions of baseball fandom: arguing that the league is manipulating the characteristics of the ball in order to change the results of games. We’re something like a third of the way through the season, now, and a lot of offensive stats are down from last year, which has a lot of people saying the league has doctored the balls to reduce the number of home runs, for whatever reason.
This specific example caught my eye because it’s unusually quantitative, thanks in large part to the vast amount of ball-tracking data that is compiled these days— basically every ball hit in play has its launch velocity and launch angle recorded using automatic video analysis. It’s also very much an introductory Newtonian physics problem: projectile motion takes up a good chunk of a week in our classes for first-year physics and engineering students, and I always use some baseball examples of exactly this form: given some launch speed and angle, how far will the ball travel?
It’s a really easy problem under the idealized conditions of first-year physics classes, where we neglect air resistance. If you’re lazy, you can even find slightly shady websites that will do exactly this problem for you: you put in the launch height, launch speed, and angle, and it will spit out the distance at which that projectile will hit the ground.
But that also makes this tweet a little puzzling when it’s offered as evidence that the league is doctoring the balls. The usual complaint is that they’ve “juiced” the balls by making them more springy, or “deadened” them by making them less springy, but either of those changes affects the speed of the ball coming off the bat. Once you specify the speed and launch angle, you’re all done— there’s only one answer, with the only wiggle room being converting from American units to metric. If the balls were “dead,” you would get fewer balls hit at 107mph and a 30 degree launch angle, but those that are would still land in the same place. For the specific numbers in the tweet, the single predicted landing position is about 675 feet away.
Of course, if you follow baseball even a little bit, you know that’s an absurdly big number. You can find claims that Babe Ruth or Mickey Mantle or somebody of that era hit a ball 600 feet, but nobody actually believes that. In the modern era of televised games and ball tracking, there are very few examples of anybody hitting a ball even 500 feet, let alone 675. It’s way bigger than the “normal” number claimed in the tweet, an average of 421 feet.
The reason here is air resistance: in the process of making its way from the bat to the outfield, the ball needs to push air molecules out of its path, and in keeping with Newton’s Laws, those molecules push back. The ball experiences a force from the air that opposes its motion, and that force increases as the speed of the ball increases: a faster-moving ball needs to push the air molecules harder to get them out of the way faster, and that means a larger reaction force.
Air resistance is a little messy, mathematically, and doesn’t lend itself to simple solutions— the factors that go into it (the size of the ball, the density of the air, the aerodynamic properties of the ball) are reasonably well known, but because it depends on the velocity of the ball, you more or less need to model it numerically. But, on the bright side, this is also a first-year-physics-student level problem, at least if you’re teaching in a way that lends itself to incorporating numerical simulations…
As it turns out, I already had a VPython program for projectiles with air resistance that I used for something else, so I modified it to simulate baseball trajectories. Here’s the code on GlowScript if you’d like to see how it works or laugh at my rudimentary coding skills, and here’s an example of the output:
The two lines represent trajectories for two balls launched at an angle of 30 degrees from a height of 1.5 meters with a speed of 48 m/s, which are the SI versions of numbers from the tweet. The white ball has no air resistance, the yellow is with air resistance in the usual intro-Newtonian-physics form; the distance in feet to the landing point is printed out below the image. You can see that air resistance makes a dramatic difference.
In this specific case, I fiddled with the parameters of the air resistance model to get a distance of 420 feet. This uses a drag coefficient (the number that describes the aerodynamic properties of the ball) of 0.25, which is a bit lower than what you usually see in textbooks, but not wildly off base. So, if you were in charge of Major League Baseball and wanted to knock this down to 380 feet by changing the properties of the ball, what would you need to do?
Well, there’s really only one thing you can do, which is to increase the drag on the ball. If you bump the drag coefficient up to between 0.32 and 0.33— that is, increase it by nearly a third— you can turn a 420-foot home run into a 380-foot pop fly. That’s kind of a big change, though, and you’d expect to see a similar effect on all the trajectories in a baseball game— outfielders trying to throw the ball in would see all their throws shortened as well. I’ve heard a bunch of chatter about batted balls supposedly not going as far, but I don’t think I’ve heard the corresponding defensive complaint— Aaron Judge saying “Man, the ball feels really sticky, and just doesn’t throw right…” or whatever.
There’s also a “Drag Dashboard” using pitch-tracking data to infer drag coefficients for individual games, and that doesn’t show a big increase for this year over the past several. (The absolute value they get is on the higher end of the range I used, but again, my model is very basic and anything involving aerodynamics is a nightmare to calculate, so I would tend to look at changes, not absolute values…)
What else could you do? Well, I added a crude model of a headwind to the code: basically artificially increasing the horizontal speed of the ball for the purposes of the air resistance calculation, as if there were a breeze coming in from the outfield. This will also tend to shorten the distance traveled, turning balls that “ought to” be home runs into long pop flies.
How much headwind do you need to turn a 420-foot homer into a 380-foot fly ball? For my crude model, between 7 and 8 m/s, which is around 17mph. And, interestingly, if you look at the replies to that original tweet, you’ll see someone claiming that the wind was 18mph coming in from the outfield. That’s implausibly good for a model as crudely coded as the one I’m using here, so I wouldn’t take it too seriously, but it shows that the scale of the effect isn’t wildly unreasonable— you can get that kind of change in flight distance without needing hurricane-force winds.
So, as I tweeted last night, I don’t think this particular clip works as evidence that MLB has, for inscrutable reasons, chosen to doctor the balls in a way that makes home runs less likely. It doesn’t mean that they haven’t, mind, but if they did, it would have to have been through some change in the drag properties of the balls, which you would expect to show up in throws in from the outfield as well as long hits, and that’s not a thing I’ve heard anybody complaining about. It’s more plausible that it was just a windy night, and the batter here was robbed by nature, not the league office.
So, you know, there’s a little old-school physics blogging for you. This was fun; I’ll have to try to do more. If you want to see whether I succeed, here’s a button:
And if you have alternative theories or criticisms of my toy model, the comments will be open:
The Oracle Park "glove" in San Francisco is 501 ft from home plate. A target for everyone, including your guy Judge.
NEEEEEEEEEEEEEERRRRRDDDDDD! No, this is really great -- I'm not saying I fully understand all of it, and certainly my replicating even the basic math would take more time than for a smart person, but I do love what amounts to the debunking of this as simply a "they are doctoring the balls again!" argument.
All I can say by way of sporting analogy is that I spent four years long and triple jumping in college dying for a gust of wind at my back that hit my back the second my foot hit the board. It never happened, reaffirming my loathing of meteorology AND physics AND my own physical limitations.