The Physics of the MLB Report on Home Run Rates

Bryce Harper is hitting home runs at an even greater rate this season than he did last season. (via slgckgc)

On May 25, Major League Baseball made public its “Report of the Committee Studying Home Run Rates in Major League Baseball”. The committee was headed by Dr. Alan Nathan, and its findings stated, “…measured changes in the drag coefficient since 2015 can explain the observed increase in home run production…”

The report leaves us with a number of questions, not the least of which is: What the heck is the “drag coefficient?”

Let’s start with the basics. The flight path of a baseball is determined by three forces. Gravity pulls the ball down toward Earth throughout its flight. The air around the ball exerts the other two forces.

The air interacts with a spinning ball, resulting in a force referred to by two different names, the lift force or the Magnus force. I’ve tried to explain the origin of this force before at The Hardball Times.

The second force exerted by the air also has two names that describe the same idea, the drag force or air resistance. This is the force the report focused on. For clarity, I will henceforth use the words drag or drag force and ignore the term air resistance.

Since the drag force slows the ball, it makes good sense that if the drag force on a fly ball drops enough, the ball is likely to go farther. So, first we’ll look into the behavior of the drag force. Once we get a grip on how the drag force works, we can move on to understanding the drag coefficient.

In principle, the way to completely understand the drag force would be to keep track of the behavior of each and every air molecule that interacts with the ball during its flight. The trouble is, for every inch the ball travels, it has to get about 10 billion-trillion air molecules to move around it. So, this microscopic approach won’t work. Instead, let’s use macroscopic reasoning.

Do you have a coffee filter handy? Off-the-wall question, I know. However, coffee filters are a great demonstration of the drag force. Drop one, and you’ll notice it falls rather slowly compared to things like your keys or a coin. The drag force on a coffee filter is a higher percentage of its weight than for your keys or a coin.

Now, let’s see if we can figure out what the drag force depends on. Think back to the last time you tried to walk through water that was chest high. I think you will agree it is harder to walk through water than air. The drag force on your body is higher in water than air. This is because water is denser than air. So, the drag force depends upon the density of the air through which the ball is flying.

When you were walking through the water, you might have noticed it was easier to walk slowly compared to quickly. So, the drag force depends upon the speed a baseball moves through the air. Another way to understand the dependence of drag upon speed is to put your hand out the window of a moving car.

Now, understand I don’t actually recommend you do this, because you’ll feel pretty foolish trying to explain how you got hurt once you get to the emergency room. However, if you have by chance performed this experiment, you would notice the drag force on your hand does indeed grow as the speed of the air passing by your hand increases. In fact, the drag grows faster than the speed of the car increases. Careful measurements reveal the drag force usually grows as the square of the speed.

Next, get two identical pieces of writing paper. Leave one flat, and wad the second one up into a ball. Drop them both at the same time, and you will notice the one you wadded up falls faster than the flat sheet. This experiment shows the drag force depends upon the surface area.

At this point, we can summarize our experiments with an equation for the drag force, Fd, in terms of the density of the air, ρ, the speed of the object moving through the air, v, and the cross-sectional area, A, as expressed in equation 1.

Where did the one-half come from? Don’t worry about it. It is the least of our problems. Remember, we want to understand the drag coefficient, which hasn’t even reared its ugly head yet. The reason? There is another thing the drag force depends upon – the shape of the object moving through the air.

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If you keep the air density, speed, and cross-sectional area the same and compare the drag force on a cube with one face moving into the air compared to a sphere, guess what? The sphere feels less drag than the cube. The sphere is more aerodynamic than a cube. We need to add something to equation 1 to account for this. It is the drag coefficient, Cd. Here is equation 2.

It turns out, the drag coefficient doesn’t just depend upon the shape of the object. Cd describes all the aerodynamic properties of the object moving through air. These get really interesting and very complicated. For example, the texture of the surface of an object is related to its aerodynamic properties, but not in a simple way.

Most of the time, you would suspect a smoother surface would be more aerodynamic, but this isn’t true for all spheres. For example, golf balls with dimples travel further than smooth golf balls. So the drag coefficient for a rough golf ball is actually less than it is for a smooth golf ball.

One would suspect, based on the golf ball, that a smooth baseball would have a higher drag coefficient than the current ball. Yet it also stands to reason that if the seam height gets tall enough, the drag coefficient would grow. So far, studies show the common-sense idea that the higher the seam, the greater the drag for baseballs. In general though, the connection between the surface roughness of the baseball and the drag coefficient is not well known.

In addition to the drag force depending upon the surface roughness, experiments show it also depends upon the speed of the ball. For low speeds, the drag force is higher than it is for high speeds. This effect is tossed into the drag coefficient as well.

Now imagine the flight of a lopsided ball. As it wobbles through its trajectory, it will interact with more air than a uniform ball. In technical terms, a lopsided ball is said to have its center of gravity offset from the true middle of the ball. This effect is lumped in with the drag coefficient, too.

In addition, the drag force on the baseball depends upon its spin. This also is included in the drag coefficient. The drag coefficient is really a catch-all for the various behaviors of the drag force that aren’t explained by equation 1. Fundamentally, all these complications are due to the fact that we really can’t keep track of the behavior of each and every air molecule as it interacts with a baseball in flight.

Now, back to the MLB report. Once the study completed its very thorough analysis of every other possible cause for more home runs, its authors discovered the only change was greater carry on fly balls caused by smaller drag forces. They checked the air density, the speed of balls off the bat, and the size (surface area) of the balls.

Since they found no change over time in these parameters, the drag coefficient is the only thing left to blame. The committee went on to conduct experiments on three potential causes of the changes in the drag coefficient – seam height, center of gravity, and surface roughness.

They found, “The average seam height in these data does not appear to correlate with the trends of yearly averages of Cd values. For example, the 2017 seam height is near a 15-year high, whereas the mean Cd is at a recent low.”

As far as the center-of -gravity issue, “These results are consistent with the aforementioned line of reasoning and suggest that differences in the ball center of gravity may be contributing to changes in ball drag. Additional work to study this effect is ongoing.”

Their experiments resulted in, “…lending some credence to the notion that surface roughness plays a role in baseball drag. This is another line of research that will be pursued in the near future.”

The MLB report made recommendations regarding ways to address the drag coefficient issue:

Recommendation 3: MLB should monitor and attempt to standardize the application of mud on the baseballs, since the surface texture of the baseballs affects drag.

Recommendation 6: MLB should continue to study the drag properties of baseballs.”

I hope you’ll forgive the lengthy exploration of the ideas behind the drag coefficient, but that’s the only way to really understand the MLB report. I hope MLB will pursue the lines of further research suggested by the committee. If not, I guess we’ll be left with only one thing to say: What a drag.


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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TheWiking
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TheWiking
There are two aspects of the report about which I’m still curious. The committee concluded that the change in the pill mold (it’s larger now) didn’t have an impact they could measure. I’m surprised by that, as I would expect that a larger pill could result in a more consistent placement of the center of gravity. I’ll be curious to see where that line of inquiry goes. The second aspect is foreign substances (FS’s). The spin advantages that a pitcher would gain from applying a FS would also likely apply to spin imparted from contact. This would be consistent with… Read more »
channelclemente
Member

I think viscosity of FS is the key variable. It’s also possible that some smart guy has either rationally or empirically stumbled on to some FS that exhibits non Newtonian flow at the angular velocities or linear velocities experienced by a baseball.

channelclemente
Member

Great article.

timprov
Member
timprov

These results are consistent with the aforementioned line of reasoning and suggest that differences in the ball center of gravity may be contributing to changes in ball drag.

Unless I’m reading this very wrong, this seems to mean that we’re seeing more home runs because we’ve gotten better at making baseballs. Drag would typically only be reduced by moving the center of gravity closer to the center of the ball, which was the goal all along.

Vil Blekaitis
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Member
Vil Blekaitis

Great article, Mr. Kagan. All of them are btw. The equations are easy to understand, even for non-math majors like me.
One line I’d probably omit and I suspect Gaylord Perry would agree: “In general though, the connection between the surface roughness of the baseball and the drag coefficient is not well known.” Gaylord certainly understood that altering the surface of a baseball alters the drag coefficient. (Even if he wouldn’t have necessarily used that term).

mgwalker
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mgwalker

Thanks for an enjoyable article.

I read the MLB report with fascination, but was left wondering why there was so little discussion of possible changes in the lift force, as might happen if there have been systematic changes in the spin of batted balls.

I noticed only one mention of batted-ball spin in the MLB report, which indicated that the authors controlled for spin. But given the lack of public data on batted-ball spin, I was puzzled by how this component got such little discussion.

Related question: why are batted-ball spin data not public?