Like many of you, I was minding my own business on Saturday, March 29, when I got a text from a well-connected friend asking me what the deal was with the new bats that the Yankees were using and whether they were responsible for all those home runs. Of course, having been preoccupied with other things, I had no idea what he was talking about. But I very quickly found out, as I have since been bombarded with questions from people I haven’t heard from in ages, as well as interview requests from the media. As I write this, a week has passed, many articles have been written, and lots of people have weighed in on these new bats. But while this article will appear rather late in the discussion, I am hopeful it will provide some new insights into the so-called torpedo bats. And as I am want to do, I will discuss what I have learned from a physics perspective.

Before getting into my quantitative analysis, I first want to discuss the torpedo bats more qualitatively, as they were presented in a recent FanGraphs article by Davy Andrews. With a beautiful image that pretty much tells the whole story, Davy shows three different regions of a typical baseball bat: the skinny handle (“total garbage”), the sweet spot zone, and the 3-4 inches at the tip (“more garbage”).

The sweet spot zone is the region where batters prefer to make contact, while the remainder of the bat is “garbage,” meaning that it contributes very little to the performance of the bat. In an ideal world, a bat would be like a golf driver, where essentially all the mass is concentrated in the head. Of course, a golfer doesn’t have to cope with 95-mph pitches that might end up anywhere in the strike zone, so that design wouldn’t work for a bat. However, it’s still true that the very tip of the bat is fairly useless, in that it rarely results in good contact. Some wood there is necessary, just to protect the outer half of the plate. But if a hitter makes contact there, better to foul it off and live for another pitch than to hit a weak grounder to the infield. So the essential idea of the torpedo bat is to remove mass from the end, where it’s not needed, and move it to the sweet spot zone, where a hitter would like to make contact.

I can’t let this moment pass without recognizing that this basic idea was conceived of long ago by Paul Kirkpatrick, a physics professor at Stanford. Quoting from his pioneering 1963 article in the American Journal of Physics, “The mass should be where the collision is expected to occur.” (Thanks to Erica Block for reminding me of this very influential paper.)

Continuing my qualitative discussion, there are two (and probably many more) generic schemes for moving mass from the barrel tip to the sweet spot. On the one hand, the fact that the mass of the bat is closer to the hands means that the “swing weight” (or, technically, the moment of inertia, or MOI, of the bat about the knob) will be reduced, resulting in a likely increase in the rotational velocity of the swing. It’s not clear without a detailed calculation whether that will result in a greater bat speed at the sweet spot, since the lever arm from the rotation axis to the sweet spot will also be lower. Nevertheless, the bat will be “quicker,” meaning that the batter can accelerate it from its initial position into the hitting zone more quickly, allowing him to wait just a bit longer before committing to swing. Moreover, the bat will be more maneuverable, making it easier for the batter to alter his swing once it has begun, as he gathers more information on the pitch location. A quicker bat is a definite advantage.

An alternate scheme is to take advantage of the mass being closer to the hands by increasing the diameter of the bat at the sweet spot, perhaps to the maximum allowable diameter of 2.61 inches. In a sense, the bat might be “swing weight neutral,” where the advantage comes not from better quickness but from a larger hitting surface. Indeed, in his book The Science of Baseball, Terry Bahill proposed something back in 2019 that looks remarkably similar to a torpedo bat. In his words:

The bat shape is important because… to maximize the vertical size of the sweet spot, the bat diameter in the region of the sweet spot should be as large as allowed by the rules. In this respect, manufacturers of maple wood bats are punishing batters: manufacturers should enlarge the bat diameter in the region of the sweet spot, and then they can reduce the diameter in other less important regions of the barrel.

The idea of removing mass in the barrel of the bat to reduce the swing weight is not a new one. For example, many big league bats are “cupped” at the end for just this purpose. Another example is corking a bat, in which a cavity is drilled in the barrel along the axis, about one inch in diameter and 12 inches deep. (Backfilling with cork serves no useful purpose other than to hide this illegal act.) Once again, the idea is to reduce the swing weight by removing weight from the barrel. Finally, one of the principal advantages of hollow aluminum bats (other than the trampoline effect) is that they inherently have a smaller swing weight than a comparably sized wood bat, resulting in the quicker-bat advantage. So while the essential idea isn’t new, the novel implementation of the idea and the ability to convince major league batters to adopt it is new, thanks to the insights and efforts of physicist Aaron Leanhardt.

These qualitative aspects of the torpedo bats were things I learned within a day or so after the Yankees’ slugfest. But I asked myself whether there might be more going on than meets the eye. Is it possible, as some have suggested, that a batter can hit the ball harder with a torpedo bat? In particular, were these novel bats the primary reason for the Yankees’ home run explosion? Or, as others have argued, is the size of the sweet spot zone larger than for a conventional bat? These are the kinds of questions that interest me and have led to the initial investigation that I will tell you about in this article.

The basic tool I will use is computer code I developed many years ago to study the physics of the ball-bat collision. As discussed in my November 2000 American Journal of Physics article, “Dynamics of the Baseball-Bat Collision,” the collision model goes beyond the rigid-bat approximation to include the important effects of bending vibrations in the bat. As discussed in the paper, in a simple one-dimensional approximation, the exit velocity (EV) of the ball from the bat can be expressed as:

EV= q*(pitch speed) + (1+q)*(bat speed)

This “master formula” is remarkably simple in that it relates EV to the pitch speed, the bat speed, and a quantity q that is called the “collision efficiency.” As the name suggests, q is a measure of how efficient the bat is at taking the incoming pitch, turning it around, and sending it out at high speed. It’s a joint property of the ball and bat, and can assume values between -1 and +1, although it’s almost never greater than about 0.2 in a game situation. All other things equal, when q is large, EV will be large. We all understand about pitch speed and bat speed, but q deserves further comment, given the important role it plays in determining EV.

The collision efficiency depends on two other quantities: the ball-bat coefficient of restitution (BBCOR) and the recoil factor r. The BBCOR is related to energy dissipation in the collision, which can occur in both the baseball and in the bending vibrations of the bat. When the BBCOR is large, there is less energy dissipated, so that the collision efficiency is larger and the EV is greater. The opposite occurs when the BBCOR is small. The recoil factor r is approximately proportional to the inverse of the MOI of the bat. A necessary consequence of Newton’s third law of mechanics is that the ball transfers some of its energy to the bat in the form of recoil. When the bat is light (low MOI), it recoils more, resulting in more energy transferred from the ball and therefore a smaller collision efficiency and a lower EV. And the opposite when the bat is heavy. But it’s important to keep in mind that the MOI also controls the bat speed. A higher MOI almost surely means that the bat won’t be swung as fast and the opposite for a lower MOI. Therefore it’s not at all obvious that reducing the MOI by moving mass from the tip of the barrel towards the knob will result in a larger EV, since the higher bat speed is at least partially offset by the lower collision efficiency. It requires a detailed calculation to determine which of the two effects “wins.” I will now describe such a detailed calculation.

The ultimate goal of the calculation is to determine the variation of EV along the barrel of the bat. A necessary first step is to determine the bat profile, by which I mean the variation in the bat diameter along its length. Rather than do my own bat design, I decided to get realistic numbers from bats actually used by a major league hitter. From an anonymous source, I was able to obtain the profile of both a conventional and a torpedo bat, both used by a certain big league hitter, allowing a side-by-side comparison of the expected performance based on my simulation. The profiles are shown in Figure 1.

While the maximum diameter is about 2.5 inches for each bat, it has shifted away from the tip and toward the sweet spot zone in a noticeable way. Both bats were 33.5 inches long and weighed 31.5 and 32.1 ounces, respectively, for the conventional and torpedo bat. These profiles can be used to find the volume of each bat, so that the density of the wood can be adjusted to reproduce the known weight, resulting in the density being about 3.3% greater for the conventional bat. Once that is fixed, the MOI can be calculated and, remarkably, it’s identical for the two bats; I suspect that that feature was a design goal rather than an accident. It’s therefore expected that these bats will be swung identically, with no particular advantage for either with regard to swing speed or bat quickness. Moreover, the recoil factor described above will also be essentially the same. As a result, it’s expected that the only factor that can contribute to differences in EV is the BBCOR.

Before embarking on a calculation, one additional unknown parameter has to be determined: the Young’s modulus Y of the wood, which is a measure of the bending stiffness of the bat. When Y is large, the bat is stiffer and harder to bend, resulting in higher vibration frequencies and less energy transferred to the bat in the form of vibrations. As a result, the BBCOR will be larger, resulting in a higher collision efficiency and a greater EV. The role that bending stiffness plays in bat performance was discussed in a talk I gave at Saberseminar in 2023. In practice, Y is easily measured, but since I don’t have the two bats in my possession, I made the simple assumption that Y was the same for the two bats.

With the bat profiles, densities, and bending stiffnesses all fixed, the collision efficiency can now be calculated at locations along the length of the bat. To determine EV requires deciding on both a fixed pitch speed and a swing speed formula. I assumed an 86 mph pitch speed, which is approximately the speed at home plate of a pitch released at 95 mph. For swing speed, I assumed that the bat is pivoted about a point on the axis of the bat but 2.5 inches off the knob, based on experimental data. One might quibble with this model, but it certainly should be good enough for comparing the two bats. The results of the calculation are shown in Figure 2.

Referring to Figure 2, here is a summary of what these calculations are telling us:

• The EV maximum is approximately the same for the two bats, although shifted by about 0.5 inch away from the tip for the torpedo bat. For convenience, I will refer to the location of the EV maximum as the sweet spot.
• Relative to the standard bat and sweet spot, the torpedo bat performs worse closer to the tip and better close to the handle. The gains on the inside far exceed the losses on the outside.
• As a result, the width of the sweet spot, which I (arbitrarily) define to be the range of location with EV exceeding 100 mph, is greater for the torpedo bat.
• EV and BBCOR both peak at the sweet spot, which is also where the vibrational energy is minimized (not shown on the graph). The collision efficiency peaks about one inch towards the handle from the sweet spot. These are general features of the ball-bat collision, not confined to the present bats.
• The variation of BBCOR with impact location is almost totally due to the vibrations in the bat. At the sweet spot, where vibrations are minimized, the BBCOR is due almost entirely to the ball alone. Once again, this is a general feature of the ball-bat collision, not confined to the present bats.
• Given that the Young’s modulus was the same for the two bats, the difference in the vibrational properties that give rise to the difference in the BBCOR must be due to the different mass distribution. A separate calculation was done for the torpedo bat in which the Young’s modulus was changed so that the vibrational frequencies were the same as for the conventional bat. The results barely changed. I wish I could say that I have a complete understanding of this feature, but I don’t, at least not yet. I’ll be working on this in the coming weeks.

Everything I have presented here is based on a calculation. These calculations have had an excellent record in the past at predicting field performance. Nevertheless, physics is an experimental science and there is no substitute for good measurements. My collaborators (Dr. Lloyd Smith of WSU and Dr. Dan Russell of PSU) and I will shortly be doing laboratory measurements that allow direct measurements of the vibrational properties of the bats (Dan) as well as high-speed impact to measure BBCOR and collision efficiency (Lloyd). Moreover, as these bats have greater use in the majors, a lot of performance data (EV, bat speed, impact location, etc.) will be gathered that will help further elucidate the issues I have addressed here. It should be a fun summer!

…

This article is dedicated to the memory of Dr. David Kagan, my friend, co-author, and sole author of the brilliant series of articles, “The Physics of…”, in The Hardball Times. He is now in Dodger heaven, undoubtedly explaining the finer points of the physics of pitches to Fernando Valenzuela.