Billy Hamilton: faster than the optimal path

Cincinnati Reds minor leaguer Billy Hamilton is making noise with his speed – he set the minor league steals record on Tuesday, surpassing Vince Coleman’s 145 (though Coleman did it in fewer games). Hamilton was on my radar for the Portland Peskies awhile ago, after he clocked a 13.8 second in the park home run. In case you’ve forgotten, the Peskies are a hypothetical team built on undervalued baseball skills: speed, bunting, defense, and even knuckleball pitching. I’m convinced that you could build a decent baseball team for very little money by focusing on these skills.

Upon seeing Hamilton’s in the park home run time, I immediately started the mental math: 360 feet is about 110 meters, but you need to bow out to turn the corners, so maybe 120-130 meters is a good guess. The world record for the 100 meter dash is about 9.6 seconds, so a world class sprinter could do that distance in about 11.5 to 12.5 seconds — in a straight line.

But those corners complicate things. They slow you down, which makes it difficult to determine the optimal path. At one extreme, you could minimize the distance by running directly to each base and cutting hard. This is surely not the fastest way. At the other extreme, you can draw out the circle that touches each base. Again, surely not the fastest, since you don’t have to make a turn at home plate.

So, what’s the optimal path? A couple of math professors worked with a student at Williams College* to try to figure it out. The path they end up with has the batter bow out at a 25 degree angle, swing even wider than the circle from first to third, and aim for home on a relatively straight line. This path is far wider than most batters run.

Not a bad effort, but I’m not buying it. In order to prove that their path is optimal, the authors had to assume that the batter has a constant total acceleration and deceleration. They used 10 ft/sec2 for the article. These assumptions generate a circling time of 16.7 seconds on their optimal path and a superhuman speed of 42 ft/sec crossing the plate (Usain Bolt has just barely eclipsed 40 ft/sec). Something doesn’t seem right.

Assumptions are never perfect. The authors note that Bolt’s initial acceleration is much faster than their model assumption, so maybe things balance out. However, I’m pretty sure they tamped down the constant acceleration to keep the final velocity from exploding, and doing that makes it hard to compare to reality. Their time (16.7 seconds) is WAY slower than the true optimum. Some dude named Evar Swanson ran the Guinness Book record at 13.3 seconds in 1932, and of course our man Hamilton has 13.8 (coasting home). They could up the acceleration to match these numbers, but the real concern is that the optimal path is incorrect due to the constant acceleration assumption.

So, I turned to my brother Evan — who studies the engineering of the human body — to figure out where they went wrong. In short, Evan says that constant acceleration gives you the wrong path and the wrong total time. First of all, maximum forward acceleration decreases as velocity increases, which keeps the baserunner in check down the third base line. More importantly, maximum lateral acceleration may actually increase as velocity increases (think about a wide receiver coming into a hard cut at full speed versus walking speed — the legs driving into the ground act like springs and translate some of the forward velocity into lateral acceleration). So, holding total acceleration constant (lateral and forward combined) is problematic.

The Williams guys suggest that velocity will increase up to first base, then dip slightly to make the turn, increase slightly until second base, decrease slightly on the way to third base, then kick up to full speed on the way home. Speed is held down in the middle because most (if not all) of the limited acceleration is being used to turn. This limitation is probably exaggerated.

There’s also the small matter of the bases themselves. My buddy Tony, a former baseball player, says that leverage off the base is extremely important, especially for a good baserunner who knows to get his outside leg on the inside of the base. Again, this allows for better cornering, suggesting that a tighter path is faster.

He also had this to say:

As a runner you are trying to set yourself up for not having to slow down. So at least from home to first you basically run as fast as you can and you certainly don’t slow down at first. I would say you are speeding up as you pass it. Likewise, you set yourself up to round second and you accelerate through the base.

Perhaps the assumptions are necessary to get a proof done, but it seems like the resulting velocity trajectory is way off from reality, which makes me very suspicious of the conclusion. Hopefully, I’ll have a chance to go out and actually time some different paths soon.

A related question: across what distance can humans attain the highest average speed? The obvious trade off is to run farther to offset the slow start, but not too far due to fatigue. Usain Bolt’s 100 and 200 meter world records (9.59 and 19.19) imply nearly identical average velocities. He actually ran a faster average velocity in a 150 meter race (14.35 seconds). In the 100, sprinters attain maximum speed between 60 and 80 meters, but the optimal distance will be the point at which current velocity drops below average velocity (which implies that running farther will drag down the average). This distance could be pretty close to the distance of the optimal baserunning path: about 115-125 meters, though a few more data points would help. Another upshot: baseball players probably aren’t traveling at top speed anymore when they reach third base and hustle home, which is one more reason to tighten up that corner.

*Williams College is known for its excellent math department. Yours truly, a Middlebury College grad, did not realize this study was by Williams guys when I looked it up, but let’s just say it provided some extra motivation.

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