The Physics of the Cutoff: Part II

Hitting the cutoff man is paramount at times. (via Dru Bloomfield)

Hitting the cutoff man is paramount at times. (via Dru Bloomfield)

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.

Arm Strength Launch Speed (mph) Back Spin (rpm)
Great 100 2,000
Average  90 1,500
Poor  80 1,000

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.

Cutoff Throws Poor Average Great
Poor 200 300 400
Average 300 200 336
Great 400 336 200

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.

Cutoff Throws Poor Average Great
Poor 3.51 3.23 2.87
Average 3.23 3.09 2.86
Great 2.87 2.86 2.76

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.

Cutoff Throws Poor Average Great
Poor 200 227 252
Average 227 200 226
Great 252 226 200

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.

Cutoff Throws Poor Average Great
Poor 4.45 4.09 3.74
Average 4.09 3.80 3.49
Great 3.74 3.49 3.24

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.

David Kagan is a physics professor at CSU Chico, and the self-proclaimed "Einstein of the National Pastime." Visit his website, Major League Physics, and follow him on Twitter @DrBaseballPhD.
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8 years ago

I agree: those tables are confusing. How about putting those data into a line graph or even a bar graph? Make the horizontal axis arm strength for either outfielders or infielders and the vertical axis distance or time. Then make three different lines or bars for each level of arm strength for the other type of player (color code them). If the arm-strength variables were continuous instead of discrete, you could even visualize distance or time as a surface in a three-dimensional graph.

Here are a few other variables to consider to extend the analysis, in no particular order:

1. Players can’t teleport, so the infielder needs to be able to reach the cutoff position by the time the outfielder is ready to throw the ball. Perhaps the infielder can continue running toward the outfielder while he throws, but if the infielder runs off of the line from the outfielder to home plate, then this may negate some of the benefit of reducing the distance of the outfielder’s throw. Also, the outfielder would have to throw low enough so that the infielder can still catch the ball. I don’t know if that would have an adverse effect on the outfielder’s throw or the infielder’s ability to catch and throw, but it might.

2. Infielders do not instantaneously catch and release the ball. The delay must be accounted for.

3. Accuracy might matter more for infielders than for outfielders. If the infielder runs farther out into the outfield grass to make his throw, this could worsen the accuracy of his throw to home plate more than is desirable. The throw from the outfielder probably doesn’t need to be quite as accurate, since the infielder can reposition himself to catch the ball (within reason). Home plate can’t move around of its own volition (though this would be rather entertaining to watch).

4. Players’ initial and throwing positions may affect the strength and accuracy of throws. Do players who have to run farther to get the ball produce weaker or less accurate throws? What about players who release the ball more quickly and whose footing is perhaps less set as a result? What about players running toward their glove side or arm side? What about outfielders running straight back or at some other angle?

I’m gonna stop now.

8 years ago

This is awesome work!

I usually call this play a ‘relay’ throw rather than a cut-off to distinguish it from the other option of stopping a throw and re-directing it elsewhere. The cut-off man, as you point out, is usually closer to home. He’s there to keep the batter/runner at first more than to help the ball along.

Conversely, the relay man is a middle infielder and he is usually positioned somewhere in the outfield. Your work gives a nice starting point for the throw from center.

The cut of the outfield grass behind second base is about 155 feet from home. So, in the case of a great arm in center and a poor one at short, the first throw would reach the infield dirt before being relayed (rarely see this). The other scenarios result in a throw either 18 feet away from the cut of the grass, or 45 feet (see this all the time).

I think your work tends to support the conventional way relays are handled, which is great in one sense and too bad in another. Great work, regardless! Thanks for sharing.

8 years ago

Jonathan’s point 4 is the key that makes the initial article a fun exercise instead of a how to. Quick max effort throws have a launch angle imprecision that weight heavily on a search for optimum positioning.

The three kinds of accuracy needed situations are:
1) good enough to be caught ( cutoff )
2) good enough to be caught with a foot on a base ( force play )
3) good enough to be caught and tagged.

Here the first throw fits #1 and if a cutoff can easily take several steps in any direction during the flight of the first throw to make its inaccuracy have minimal impact on total time until tag.

But the second throw is in #3 where even minor inaccuracies can turn it into uncatchable for the catcher, but more importantly left right inaccuracies greater than an arm’s length can significantly delay the application of the tag.

Cases of these appeared in the last two world series, as Duda’s hasty throw was easily in time to get Hosmer at the plate from 90 feet, but was angularly inaccurate to make timing immaterial. And then the previous year Gordon being stopped at third because the coach estimated Crawford’s deep relay’s chance of an angular inaccuracy was less than Salvador Perez getting a two out hit to score a runner on third.

8 years ago

David, I responded here:

As for the tables: simply have the better arm on one axis and the worse arm on the other would remove the confusion.

Richard Ray
8 years ago

At last some physics I can play with! I think the best representation would be a x-y graph with x being the locations of the two throwers and home plate and y the path of the two throws, end to end. This could easily be set up interactively in Excel with a slider tool to set the players throwing strengths and necessary positions. As a shortstop I agree with the other comments about all the variability; that’s why you have to physically play the game. Some other considerations 1. The relay man should always have a better relative performance (closer to his maximum throw because he can set up better than the out fielder. 2. Relay accuracy is far more critical (within perhaps a 3ft x 3ft target). 3. Outfielder and relay release cycle time are also critical (cycle time = time from first touch of ball to throw release) and can vary. Of course pros are very good at this, but even they vary widely. Thanks for the fun!

Stephen S Dudas
8 years ago

I think your on to something that could alter some strategy, but as the comments suggest there are a ton of other factors that influence this type of play. It seems like for now we have to sit back and marvel at the professional players and managers ability to determine this strategy without any hard math.

8 years ago

Consider the human element(vectors aside) in this scenario especially when the runner is relying on the 3rd base coach. If the outfielder’s throw is quick(w/o a crow hop unless the ball off the wall & leads him into a shuffle) & it is accurate, most 3rd base coaches will hold the runner. The exception would be with 2 outs & deep in the batting order(Scoring a runner). Another exception would be if the relay man out plays his arm strength.(Is positioned too far away from 3rd or home) The coaches are pretty much dialed in on the defensive players & make their decisions on the trajectory & accuracy of the 1st throw. Probably adds value to the 1st throw being a little shorter in order to get the ball into the relay man’s hands sooner.

7 years ago

It is usual that anyone can get this.