(*When I created this site, I briefly hoped that MLB baseball would soon be back in Montreal. Putting out French content thus seemed like a timely thing to do. Looks like MLB won’t be back anytime soon, so while waiting for hope to return I might as well write for a larger audience…)
The news that MLB will have humidors in every park this season triggered a lot of fan questions about what this exactly means for a given team performance. Basically, the reason why this matter is that as a ball’s humidity increases, its bounciness decreases a bit, leading to a lower exit velocity, which translates to a lower home run probability (for a decrease in humidity, you obviously get the reverse). But it is not just about home runs, since good things also usually happen the harder a ball is hit. So, the question on everyone’s mind is how much of an impact this will have in city X which has an average relative humidity Y. We already know that the effect was quite substantial in Denver since the home runs hit per game went down when they started storing balls in a humidor at a constant 50 percent relative humidity, as opposed to the more typical 30 percent humidity.
But what about the effect on my favorite team? Can we predict quantitatively the effect of the humidor? It turns out that the subject has already been scientifically investigated quite thoroughly. Great explanations on the impact of humidors can also be found here and here so I won’t bother repeating everything.
Instead, what I am going to do in the following is to take a look at some of the assumptions that go into those calculations and provide a tool (link up here below the title) that allows you to rapidly estimate the change in exit velocity as you change any variables (as a physicist, I enjoy very much going over all the equations that can be found here and here, but I imagine that this is not everyone’s cup of tea to make all the calculation themselves, hence the user-friendly calculator).
As mentioned above, humidity level affects the bounciness of the ball, which in turn affect the exit velocity (sometimes referred to as Batted Ball Speed, or BBS). All in all, for a contact near the sweet spot, the exit velocity (EV) is given by the following simple formula (see also this):
EV = q*vball + (1+q)*vbat
where vbat is the bat speed at the point of contact, vball is the velocity of the ball at impact, and q is the so-called « collision efficiency » that depends on the properties of the bat and the ball-bat coefficient of restitution as follow:
q = (e + r)/(1+ r)
where e is the ball-bat coefficient of restitution and r is the bat recoil factor. The latter is defined as the ratio of the mass of the ball to the effective mass of the bat (kind of a way to only consider the part of the bat that matters during the brief instant that the ball is in contact with the bat):
r = m/Meff
That’s it, this is all you need to calculate the exit velocity after an impact near the sweet spot. Now let’s look how some of these variables behave with a change in humidity. Humidity affects both the coefficient of restitution and the mass of the ball in ways that has been quantified in this paper.
Let’s start with the easy one, the mass of the ball. In the study mentioned above, it was found that the mass of the ball increases linearly by 0.09 oz (2.55g) for each 10% increase in relative humidity. Although it is not the biggest factor, an increased ball mass does diminish the collision efficiency (and thus the exit velocity) a tiny bit and the calculator takes that into account. Note that the mass of the ball must be between 5 and 5+1⁄4 oz. by regulation. Assuming 5.125 oz. at 50% relative humidity (the middle value), the mass already goes outside the regulation requirements with humidity below 36% or above 64%. However, it may take several days, if not weeks, for the water in the air (humidity) to enter or leave the ball. Thus, the real water content of a ball may depend on the prior humidity fluctuation that it recently experienced. For that reason, it is difficult to assess how balls stored in humidors really compare, in terms of humidity, to those that arrived in boxes a few days earlier. Anyhow, assuming that somehow the balls have reached equilibrium with the average local humidity value, the calculator allows you to assess quickly the effect of humidity.
What about the coefficient of restitution? The coefficient of restitution (or COR) is defined as the ratio of the final to initial relative speed between two objects after they collide. It is a measure of the energy that is dissipated during impact (the fraction of the energy that is lost is given by 1 – COR2). The rules of baseball state that a ball shot at 85 ft/s at a wall of northern white ash must rebound with a speed of 54.6 ± 3.2% of the initial speed (i.e. the COR for an impact at 58 mph with a wall must be between 0.514 and 0.578). You may notice that 58 mph is far from gameplay situations where the relative velocity between the bat and the ball can be up to 170 mph! (a 100 mph fastball reaches home plate at about 90mph, impacting a bat moving at over 80 mph for an elite batter, resulting in a 170 mph relative speed). Moreover, those impacts are with a cylindrical piece of wood, not a flat wall. Measurements for impact on fixed cylindrical surfaces do not show a significant difference between the COR and cylindrical COR (CCOR), at least for relative velocities below 120 mph. What about higher velocities? I have no idea, but it seems logical to me that for more violent impacts, the ball deformation (and thus energy dissipation) might not be exactly the same on a wall vs a cylinder, especially since the center pill in a ball that is compressed by almost half of its diameter will most probably make its presence felt more strongly than in the case of a gentle 60 mph impact, where the compression is only a fraction of an inch. But without data, I can unfortunately only speculate. Also, a fixed cylinder is not the same as a baseball bat. Baseball bat will vibrate differently depending on the location of the impact point, thus affecting the value of rebound velocity (and thus the measured ball-bat COR). But for impact near the sweet spot, vibrations are minimized, and it is found that CCOR and ball-bat COR are very similar at 60 mph. For simplicity, I am simply going to assume that the COR, CCOR and ball-bat COR are not too different from one another and see where that leads us in terms of exit velocity calculations.
So, there are now two questions remaining to answer: 1) How does the COR change with relative velocity (pitch speed + bat speed)? 2) How does the COR change with relative humidity? To answer the first question, we can extrapolate the measurements shown in Figure 2 of Nathan et al. 2011. At 60 mph, we can see that the COR is about 0.55, going down to about 0.469 at 120 mph, with intermediate value of about 0.52 at 90 mph. In that regime, the COR thus appear to decrease linearly according to the relation COR = -1.35×10-3*vrel + 0.631, where vrel = vball + vbat . Extrapolating such a relation to vrel = 170 mph gives a COR of only 0.402. However, the exit velocity of a 90 mph ball blasted by a 80 mph bat would then be predicted to be barely above 100 mph, a value that makes no sense given that we do sometimes record exit velocities in the 115-120 mph range. In fact, using the formula above, producing an exit velocity of 115 mph with a 90 mph pitch speed and a 80 mph bat speed requires a collision efficiency of 0.206. For that to happen, the coefficient of restitution must be well over 0.45 at those high relative velocities (assuming a realistic value of Meff), indicating that the linear relation above must break down for velocities not much greater than 120 mph. It is unfortunate (and surprising) that measurements at higher velocities are nowhere to be found (if you are aware of such measurements, please let me know). In order to obtain exit velocities in the right ballpark, and until I find reliable measurements, I will simply assume that the COR remains constant to the 120 mph value (0.469) for higher vrel.
We can now turn to the question of how the COR changes with relative humidity. Nathan et al. 2011 show in Figure 4 that the CCOR varies linearly by about 0.0122 for each 10% change in relative humidity. Note that those measurements were carried out at 60 mph. Would the same trend be present at gameplay velocities? Possibly, I have no reason to believe it would not. So, in the absence of contrary evidence, I am going to simply assume that the fractional change of the COR with humidity at different speeds is the same as for 60 mph (this is what was done in the Report of the Committee Studying Home Run Rates in Major League Baseball). Of course, as already mentioned above, we never really know exactly what is the humidity content of a ball that is not stored in a humidor. The only thing we can do is make an educated guess.
So what does all this tells us? First, everything is a bit uncertain. More data would be nice. Nevertheless, we can be confident that the « master » formula above provides exit velocities that are not too far from reality. Refinements are always possible, but everything should be ok within a few mph at worst. Second, for the typical vball and vbat that we have at MLB level, we can estimate from the calculator above that each 5% away from 50% relative humidity will affect the exit velocity by about 1 mph. That may not seem a lot, but that translates to about 4 or 5 feet depending on the exit velocity, launch angle, and spin rate (see my trajectory simulator), enough to change a long fly ball at the warning track into a home run.
Good write-up. I absolutely love this site. Stick with it!