The Physics of the Cutoff: Part II

If you haven’t read The Boys of Summer by Roger Kahn, you should be summarily flogged. So before I have time to deal out punishments, get a copy. It will fire you up for the approach of Opening Day.

In the book, Kahn tells the story of his childhood in Brooklyn, the history of the Dodgers up until their 1955 World Series victory, and the lives of some of the ballplayers after they left the game. He often expressed frustration with his math class, “If vector analysis was beyond me, I could still watch a ball game.”

It turns out Kahn initially missed the wondrous connection between vector analysis and baseball. As an example, consider the thrilling moments as a runner starting at first tries to score on a double to the center field fence.

The outfielder picks up the ball on the warning track then whirls and throws in a single motion. The cutoff man snags the throw, pirouettes and fires home. All the while, the runner is turning third and heading for the plate. He goes into his slide just as the throw arrives.

It’s vectors – all vectors. One vector describes the launch angle and speed of the outfielder’s throw. Another represents the angle and speed of the cutoff man’s relay. Yet another vector describes the runner sliding into home. Later in the book, Kahn comes to this realization, “Baseball was simply a point where vectors converge.”

In April, 2014, I wrote “The Physics of the Cutoff”, which touched off a lively discussion. The point of the article was to compare the time for a single throw to the plate with that of two throws. Here, I want to address a different question; Where is the right place for the cutoff man to position himself to minimize the time for the two throws to get to the plate?

Now I am sure our readers have lots of good baseball strategy-based reasons for the proper location of the cutoff man. I would know nothing about that since I am just a physics professor. I only have something to say about how vectors might predict the best position to minimize the time for the ball to get to home plate. On the other hand, that’s why there is a comments section, so have at it!

As in the previous article, I’ll begin by using the kinematic equations you may have learned in high school. These equations describe the trajectory of the ball based upon its initial launch vector. They assume the only force on the ball is gravity. That is, they ignore any force air exerts on the ball.

While I could write out complicated mathematical solutions so you could plug in any initial launch speed for each throw, it will be simpler to just choose three possible arm strengths for each fielder and go from there. Then I can present results for each of the nine combinations of arm strengths of the two fielders. Here are my definitions for each of the three arms we’ll examine.

I’ll assume the total distance of the two throws combined is 400 feet. The table below shows the length of each of the two throws for each of the nine combinations of arm strength.

I find the table a bit confusing, but I couldn’t think of any other way to present the results. You’ll notice that if the outfielder and the cutoff man have equal arms, then they should split the 400 feet evenly, each throwing 200 feet as shown down the diagonal of the table.

If the outfielder has an average arm and the infielder has a poor arm, then the outfielder should throw the ball 300 feet, leaving the last 100 feet for the cutoff. That makes some sense; the fielder with the better arm should throw the ball further.

However, there is some goofiness in the table. If the cutoff man has a great arm and the outfielder has a poor arm, somehow the cutoff man should throw the entire 400 feet. That’s the problem with solely taking a scientific approach without the insight from baseball strategy. Goofiness ensues.

The total time for both throws is shown in the table below. Note this total time doesn’t include the time required for the cutoff to catch, turn, and throw the ball, because that shouldn’t vary with arm strength. Clearly, two poor arms take demonstrably longer than two great arms.

Maybe including the effects of the air on the throws can reduce some of the goofiness. As in the previous paper, I will use Alan Nathan’s Trajectory Calculator to deal with the effects of air drag and the Magnus force.

The optimal distances for the throws are shown in the table below. Again we see the obvious result that when both fielders have the same arm strength, they should split the distance, each throwing 200 feet. Also, the fielder with the better arm should throw the larger distance.

These distances are shorter than the air distances. This makes sense because the ball’s speed drops during its flight due to the air drag. So the total time will be smaller if the ball is cut off sooner and given additional speed by the second throw.

The goofiness of expecting the cutoff man with the great arm to throw the ball all 400 feet is now resolved. The poor armed outfielder should throw the ball 148 feet, leaving the last 252 feet for the cut off man. Still, in an actual game, I don’t ever recall seeing the cutoff man that far from the plate, probably due to strategic reasons.

The total times for the two throws are listed in the table below. They are all noticeably longer than the imaginary throws unaffected by the air. Again, better pairs of arms result in shorter as expected.

I would have much preferred to present some lovely formula for placing the cutoff man in the correct location to minimize the time. However, just because, “Baseball was simply a point where vectors converge,” doesn’t guarantee the mathematics will be easy.