ODE To A VogelBomb

There will be graphs.

We all know how the story goes. Hollywood has, for better or for worse, trained us on this point. The good guys meet setback after setback. Perhaps they are freedom fighters, suffering under the cruel oppression of a malevolent empire bent on total domination. They are few in number, low on resources and opportunities to strike their blow for freedom. A plan emerges to take that first, faltering step towards victory, a step that brings with it the risk of total and complete failure. All seems lost when, out of the blue, the miraculous happens:

Daniel Vogelbach became just the 3rd player ever to hit a ball in the right field upper deck in Seattle pic.twitter.com/p8nMedgp3k

— Baseball Bros (@BaseballBros) May 28, 2019

What, did you think I was talking about Star Wars? Sheesh, people, this is a Mariners blog. It's like, Star Wars isn't even relevant to anything happening in baseball land-

Star Wars is taking over @Mariners baseball both at the ballpark and tonight on the ROOT SPORTS Pregame show at 6:30 PM PT!

RETWEET & FOLLOW to be entered to win a FREE Han Seago bobblehead! Five winners will be randomly selected and announced on Monday, 6/3. #WHEREiROOT pic.twitter.com/zdqAKOqbs7

— ROOT SPORTS™ | NW (@ROOTSPORTS_NW) May 31, 2019

Okay, point taken.

Nevertheless, we are talking the VogelBomb, the dinger heard 'round the world, the ball that was hit so hard that its family is receiving compensation for the poor ball's untimely demise. There is no doubt that this is one of the longest home runs ever hit in the Mariners universe, canon and fan-fiction.

I know Statcast isn’t working tonight so I will personally take an enormous ruler to the upper deck of T-Mobile Park and measure that Vogelbach tater the old fashioned way

— Cespedes Family BBQ (@CespedesBBQ) May 28, 2019

Seriously? If we were continuing the Star Wars metaphor, that would be Luke Skywalker lining up to drop his torpedo down the ventilation shaft, only to discover that the Death Star had already blown itself up. How does a self-respecting Rebellion proceed...I mean, is there any way we can possibly know anything about that smash, short of enormous rulering?

Well, there is math.

Revenge of the Nerds

LLer henryv, in a post during the 5/30 Open Game Thread, estimated that the ball was at 75 ft elevation at a (horizontal) distance of 375 ft from home plate. We now accept these numbers as gospel. The question becomes...what do we make of them?

It's not necessarily a trivial task. We've been trained by Statcast to think of being given exit velocity, launch angles, etc., as given, and using these and what physics we can draw on to estimate the trajectory of the ball. In this scenario, we know where the ball died, and we know where the ball landed. But we don't know much about how it got there.

Any self-respecting first year physics student would, at this point, have their hand in the air, insisting that we draw a free-body diagram.

Here you go, nerds!

Of course, like a good first year student, we've neglected some very important features. The only force acting on this baseball in flight (can't you tell it's a baseball in flight?) is that of gravity. This is common in scenarios where we're trying to avoid excessive calculus and so on. And, given how often the 2019 Mariners have completely sucked the air out of the building with their on-field travails, it feels entirely appropriate to begin by pretending we're working in a vacuum. In doing so, we get the familiar kinematic equations

Nerd bait!

No air resistance makes life easy! Above, the v's are the initial speed in the x and y directions (depending on their subscript), y0 is the height of the ball when it was hit, t is time, and g is the acceleration of gravity. With some trig, we can relate these to the launch angle and exit velocity. Now, since we don't know how (and are way too lazy to find) how long it took this ball to reach its final resting place, we eliminate time from these equations (also known as Gearrinizing), include our hard earned trig work and get

More nerd bait!

The point of this is not to bore you with algebra; no more today, I promise. The point is that we have values for x and y; that's our landing point 375 and 75, which we accept as gospel. In other words, we now have a relation between exit velocity and launch angle for those trajectories that start where the ball was hit, and end where the ball was landed.

We should probably graph that. By inspecting Vogey's launch angles from this season, we estimate that launch angles from 20 to 34 degrees look promising. If we graph over that domain, here's what we get:

See, I promised a graph and here one is!

(Exit velos on the vertical, measured in mph.) From what we've seen of exit velocities from the Vogelbat and from those of other power hitters (so, say, less than 115 mph), this suggest a launch angle of somewhere between about 24 and 34 degrees.

But this is boring. For one, both teams would've long since asphyxiated from the lack of, you know, air. Seriously, though, air resistance plays multiple roles in impacting the trajectory of a home run baseball: the expected role of drag, slowing the baseball down, and surprisingly, lift.

A New Spin On Things

Our free body diagram really should look like this:

Now including lift and drag!

This is itself a bit simplified. Drag is a force that is applied in the opposite direction to that of velocity. The lift, or Magnus force, is far more interesting...and far more complicated. What is important for our work is that the Magnus force is perpendicular to both the direction of travel and to the axis of rotation of the baseball. With detailed Statcast information, we would have detailed data to use in this computation. Of course, if we had Statcast data for the VogelBomb, we wouldn't be working on this in the first place. This site has a nice, elementary discussion about drag and lift on a thrown baseball. The same principles apply to batted baseballs, as well. See also this paper which discusses more aspects of the coefficients used in these computations, not to mention this Excel sheet from Tom Tango which includes a very detailed computation.

We will perform a simpler computation, by assuming drag and lift coefficients of 0.4 and 0.3, respectively, while bearing in mind that both are functions of linear velocity, spin rate, atmospheric pressure, humidity, and so on. We also assume that the axis of rotation of the batted ball is parallel to the surface of the playing field. This isn't realistic, but it simplifies our calculation a bit.

Yes, our calculation. How do we accomplish this? The scenario here is a bit more complex than in our ideal, airless T-Mobile Park. Recall, our laziness lead us to not seek how long it took the ball to reach its destination, and instead use the fact that we know the start and end points of the actual path to compute some data about the possible trajectories. Unfortunately, when we introduced friction into the mix, we lost the ability to Gearrinize our computation (because we can't write a nice formula for distance from home plate as a function of time, basically.)

Not to worry. Imagine we have a virtual Vogelbach, a Virtualbach, if you will. Virtualbach can hit the ball however we ask him to. For instance, we can tell Virtualbach, "I'd like you to hit these pitches all with a launch angle of 20 degrees." Virtualbach can comply. But we can do even better. We tell Virtualbach, "Okay, that's great, but can you hit one with an exit velocity of 140 mph?" Again, Virtualbach complies, and we borrow a really big ruler from the Cespedes Family BBQ to measure how high in the air it was at 375 feet from home plate. We notice that it's well over 75 feet high. Emboldened, we ask Virtualbach to do the same, but with an exit velocity of 80 mph. We do the same trick with the ruler, although this time, the ball is well below 75 feet high (if indeed it even makes it to the fence.) Vogelbach would wonder why we asked him to do such a silly thing: Virtualbach complies.

Our next step is the kicker: noting that the average of 140 mph and 80 mph is 110 mph, we ask Virtualbach to hit a ball with launch angle 20 degrees, exit velocity 110 mph. Virtualbach, as expected, complies. In measuring the height at 375 for this hit, we now make a decision. If we discover that 110 mph puts us over the 75 ft threshold at distance, our next step will be to average 110 mph and 80 mph; if below, we will average 140 mph and 110 mph. We will then ask Virtualbach to hit a ball with that new exit velocity at 20 degrees. Virtualbach will comply. We will obtain a new measurement on height, where we will make an entirely analogous decision on which bound to replace. And, we can keep doing this until we're satisfied we're close to our stated goal of 75 feet in height at 375 feet away. Virtualbach doesn't care; he was literally made to hit the ball the way we want him to.

But we don't stop there. Next, we go to a 21 degree launch angle, and repeat the above process. Then to 22, 23, and so on, up to 34 degrees. Remember, we picked 20 to 34 degrees based on looking at Vogey's launch angles for hits at Baseball Savant, so we're fairly confident a good solution will be found within these bounds. The above described process is an example of a shooting method for solving a boundary value problem associated with an ordinary differential equation, or ODE. (Yes, the title of this post was math humor.) And Virtualbach, rather than being an extraordinarily precise power hitter, is a numerical technique for solving ODEs called the 4th order explicit Runge-Kutta method. You may now forget this.

But, again, you were promised a graph.

This just feels more right.

We've seen Vogey hit with exit velocity at around 115 mph (a histogram of his exit velocities is fun to see!) and leads us to a perfectly reasonable guess that his VogelBomb had a launch angle at around 29 degrees, and an exit velocity around 114 mph.

As to the trajectory itself?

It went a ways, assuming no stands.

The estimated distance, using the above computations, was 440 feet.

So, What Did We Learn?

First, dealing with drag and Magnus forces is complicated, enough so that we weren't comfortable including anything other than a simplified version, as discussed above. From what we can tell, Tango's spreadsheet (linked to above) computes the coefficients for drag and lift by taking some experimental data and curve-fitting that data to spin rate, which is itself a function of both linear and angular velocity.

Second, we learned how neat it would be to have a team full of Virtualbach's.

Finally, we learned that, as enjoyable as we found this exercise, that it does not replace the sheer awe at watching that ball climb and climb and probably destroy a seat up on the third deck. A season like this one, requires feats like that one, to get us and the players through to the other side.