Wednesday, April 17, 2024
Math behind Vertical Movement
- This is a Kodai Senga curveball, thrown in the dirt, at 63 mph.
- This is a Kodai Senga curveball, thrown in the heart of the plate, at 73 mph.
This chart shows the trajectory of these two pitches, as it travels through SPACE (click to embiggen).
Now, let me show you the height location of these two pitches, as it travels through TIME.
As you can see, these two pitches overlap a great deal.
Let's talk about vertical movement. The traditional way to measurement movement is to draw a tangent line from some point (release for example) to the plate, and compare the difference to its actual location. You can see an example here.
In this particular case, the faster curveball would have its tangent line go from release to about 7 feet above the ground. We compare that to its actual location at the plate (about 2 feet), and that difference (about 5 feet) is its vertical drop. This pitch had a vertical drop of 63 inches.
We can break it up into two components:
- the effect of gravity
- everything else
Gravity is easy enough to figure, if you remember your physics kinematic equation: one-half acceleration time-squared. At 0.514 seconds, we get 0.5 x 32.174 x 0.514 x 0.514 = 4.25 feet, or 51 inches. So, 51 inches due to gravity and 12 inches due to everything else. That 12 inches is what we call the Induced Vertical Break (IVB). (Though we really should account for the effect of drag too.)
The slower curveball would have its tangent line go also from about 7 feet to just below the ground, for a total vertical drop of 93 inches. How does that split into gravity and everything else? You might think gravity is 51 inches, but no. Gravity is not based on distance, but time. At 0.610 seconds, that means gravity pulled this pitch down by 72 inches. The remaining, the Induced Vertical Break (IVB), is 21 inches.
Now, how is it possible that two pitches, thrown with similar spin and trajectory can have two such widely different IVB? Time. You see, we are measuring the Induced Vertical Break not with a common time frame, but with a common distance frame. While both pitches travelled the same amount of distance, the slower one took much longer to get there. So, not only does gravity have more time on pulling the ball down, but the spin of the ball also has more time to push the ball down.
What if we wanted to separate that everything-else, that IVB, into both a spin component and time component, could we do that? Sure thing. First, we have to figure out how much of a time frame we are interested in. I will suggest, mostly for illustrative purposes, that we'll measure the downward movement over a span of 0.3 seconds.
When we do that, the effect of gravity is now identical for both pitches: 17 inches. We've been able to effectively neutralize gravity. And this is true not only for these two curveballs, but for any pitch, no matter how fast or slow. Gravity will always pull a ball down by 17 inches over a 300 msec time frame. In addition, the vertical drop of the two Senga pitches is also identical at 22 inches. These two balls spin the same, so given the same amount of TIME (not distance) they will also have the same spin-effect.
This is how both pitches look broken down into components, all numbers in inches:
63: vertical drop over full flight
- 22: vertical drop over 300 msec
- 17: gravity over 300 msec
- 5: induced over 300 msec
- 41: vertical drop over remaining flight
- 34: gravity over remaining flight
- 7: induced over remaining flight
93: vertical drop over full flight
- 22: vertical drop over 300 msec
- 17: gravity over 300 msec
- 5: induced over 300 msec
- 71: vertical drop over remaining flight
- 55: gravity over remaining flight
- 16: induced over remaining flight
Is there a reason to prefer 300 msec over the entire flight? Imagine for example trying to measure IVB on a curveball throw from the catcher to 2B. Because of that amount of distance (or time), that throw would have a massive amount of vertical drop, a large portion would be gravity, but also a good amount would be IVB. Even though the spin of the ball would be no different than throwing from the mound. A batter does not need to worry about the entire trajectory of the pitch, just a small portion of the flight, inside the decision-making zone. To that end, we don't need to worry about measuring everything from release or from a common distance. The decision-making zone just requires a common TIME frame.
It would therefore become a decision point for the analyst and user as to what you want to do with IVB, how you want to compare two different pitches. Are you trying to isolate the effect of the spin of the ball? Or, are you trying to include the speed of the throw as well? What is it that you are trying to isolate?
From my standpoint, we can actually isolate everything. And then the user is free to include or exclude whatever components they want.
***
A few extra math notes. You can estimate the effect of gravity using pitch speed as (523 / Speed) ^ 2
A 63 mph pitch for example would be (523 / 63) ^ 2 = 69 inches (in the above pitch we calculated 72).
And 73 mph pitch: (523 / 73) ^ 2 = 51 inches (same number as we calculated above).
This shorthand is useful especially for those who don't have the time-to-plate handy, but do we have their pitch speed handy.
***
The total amount of break is proportional to the square of the time. In the above two pitches, one took 610 msec to get to the plate, while the other was 514. Divide the two, and we get 1.19. Square it to get to 1.41. Remember how the total vertical drop for the two pitches was 93 and 63 inches? 93/63 is 1.48. We're off by a few inches, mostly because these two pitches are not identical.
We can also look at the 300 msec compared to the full flight of 514 msec. Since 514 msec pitch had a total vertical drop of 63 inches, we'd therefore expect 300 msec to be 300/514 or .584, which we square to .34. And 63 x .34 is 22 inches, the earlier number we calculated.
Similarly, the other pitch is 300/610, or .492 squared, or .24. Which we multiply by 93 to give us 22 inches, the same earlier number we calculated.
***
Anyway, I hope all this shed some light on the Total Vertical Drop and the Induced Vertical Break.
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