Alan Nathan is Professor Emeritus of Physics at the University of Illinois at Urbana-Champaign. After a long career doing things like measuring the electric and magnetic polarizabilities of the proton and studying the quark structure of nucleons, he now devotes his time and effort to the physics of baseball. He maintains an oft-visited website devoted to that subject:

When I woke up on Wednesday morning and checked my overnight Twitter timeline, I found considerable buzz about an incredible throw made by Oakland A’s left fielder Yoenis Cespedes on Tuesday night.

With the game between the A’s and the Angels tied 1-1 and Howie Kendrick on first with one out in the bottom of the eighth, Mike Trout hit a fly ball to left field that fell in for a hit. When Cespedes misplayed the bounce and the ball rolled into foul territory, Kendrick kept on running. Cespedes eventually fielded the ball, over 300 feet from home plate, then uncorked a perfect strike on the fly to nab Kendrick at home plate. Prompted by several calls on the Twittersphere for a “physics analysis” of this amazing throw, I decided to forget about what I had been planning to do concentrate on this analysis instead. So, here we go.

What I wanted to do was come up with a way to determine the full trajectory of the throw. To do that, I need two critical pieces of information: How far did the ball travel (D), and how long was it in the air (T). To answer both questions, I replayed the video over and over again until I was satisfied that I had a good estimate of both D and T.

First, we know that the distance to the left-field foul pole in Anaheim is 340 ft. I estimate from the video that Cespedes was about 20 ft from the foul pole and just inside the LF line when he unloaded the ball. I also estimate that the ball was caught by A’s catcher Derek Norris about 2 ft from the corner of home plate. Thus I find that D=318 ft. Next, I used several different camera angles to time the throw with a stopwatch. I found amazing consistency with that process, arriving at T=3.17 sec. Now, I should point out that ESPN reported somewhat different numbers, D=300 ft and T=2.78 sec. I am pretty confident about the T, and perhaps less confident about the D, since I had to estimate the distance from the foul pole. But ESPN had to do that also. In any case, I’ll proceed with my numbers and comment later on doing the same analysis with the ESPN numbers.

The next step is to use my “trajectory calculator,” a general-purpose tool I have created to do all kinds of trajectory calculations for baseball and softball. While I normally use it for pitched or batted balls, there’s no reason why it can’t also be used for thrown balls, so that’s what I did.

First, I made certain assumptions about the drag coefficient and the amount of backspin on the ball. The specific assumptions are not terribly important, since the final result is not all that sensitive to reasonable changes in those assumptions. Based on the video, I assumed that the throw was released from a height of 6 ft and was caught at a height of 5 ft. I then simply adjusted the release speed and vertical launch angle to make the trajectory be 5 ft off the ground and 318 from release after 3.17 sec. I find that the ball was released at a speed of 97-99 mph and at a launch angle of 12-14 degrees. My best estimate of the trajectory is shown in the plot below. By the way, had I used the ESPN numbers for D and T, the initial speed and angle would be 99 mph and 10.5 degrees, which are not really all that different from my results. So, there seems to be no doubt that the release speed was in excess of 95 mph and the launch angle was quite low, in the 10-14 degree range.

Having satisfied myself that I had a good model for the full trajectory, I then asked another question: How accurate did Cespedes have to be to nab the runner at home plate? The replay shows that there was very little margin for error. How does that translate in the accuracy of release?

I used the trajectory calculator to ask what would be the effect of changing either the horizontal or vertical release angles by ±1 degree. I found that a ±1 degree change in horizontal angle would lead to a horizontal deflection of about ±6 ft at home plate, probably making it impossible to nail the runner. I found that at ±1 degree change in the vertical angle would change the height of the ball at home plate by ±5 ft, meaning the ball would have hit the ground just in front of the plate or nearly gone over the catcher’s head. So we can safely conclude that Cespedes’s margin of error was less than (but comparable to) ±1 degree in each direction.

To throw the ball that hard and that quickly (after all, he didn’t have time to “aim”) with that accuracy is truly an amazing feat. Everyone who has seen the throw knows that already, but now we’ve quantified exactly how amazing it was.