The Physics of the Cutoff

The runner at second has a modest lead as the batter strokes a single through the right side.  The hard-charging outfielder scoops up the ball and unleashes a cannon shot.  The runner has turned third and is digging for home.  A smile comes to the face of the batter as he rounds first to see the throw sail over the cutoff man’s head.  He knows when the play ends, he’ll be in scoring position.

If the runner is out at the plate, it was a great defensive play.  However, if he scores, suspicious eyes turn toward the right fielder (yes, I’m talking about a certain Dodgers outfielder).  The defensive judgment of the players aside, let’s examine the costs and benefits of hitting the cutoff man from the point of view of basic physics.

Let’s suppose the outfielder releases the ball 270 feet from home plate as the cutoff man stands directly between the fielder and the plate 90 feet from home.  One way to look at the issue is to compare the time it takes for the ball to travel directly from the outfielder to home to the total time for the throw to the cutoff plus the time for the cutoff to throw home.

If we only worry about the effect of gravity on the ball, we can find these times with the kinematic equations.  You know, the high school physics you’ve forgotten.  Check your mental attic–-it’s somewhere between the theme of your senior prom and the name of that goofy English teacher with the funny glasses.

Let’s assume both the outfielder and the cutoff man can throw a ball at 90 mph.  I’ll spare you all the equations and arithmetic.  The results are shown in the table below.

Throw From Throw To Launch Angle Time
outfield home

14.9˚

2.12s
outfield cutoff

9.65˚

1.383s
cutoff home

4.76˚

0.684s

The total time for the one throw is 2.12 seconds, while the combined time for the two throws is 2.07 seconds.  It might surprise you that the time for the two throws is actually less than for the one throw home.  This is due to the higher launch angle required to get the ball home, causing the ball to travel a greater distance through the air.

Don’t get too worried about it because the difference of 0.05 seconds is far less than the time required for the cut-off man to catch the ball, turn, and make the throw home.  So, the throw directly home will be slightly quicker.  This explains why the cutoff man lets the ball go through to make the play at the plate.

The above discussion assumes that the ball and the air don’t interact.  From your knowledge of the behavior of pitched baseballs, you know air plays a critical role in the behavior of a thrown ball.  This interaction is well beyond high school physics, so again, I’ll leave out all the math.

Depending upon your level of interest, you can investigate different throws using Alan Nathan’s Trajectory Calculator or, if for some crazy reason you want to understand the detailed physics, you can read his paper.  Let’s assume the outfielder can put the same backspin on the ball as a pitcher with a good fastball, say 2000 rpm.  Here are the results.

Throw From Throw To Launch Angle Time
outfield home 11.6˚ 2.75s
outfield cutoff 5.65˚ 1.646s
cutoff home 2.15˚ 0.743s

You can see the launch angles are noticeably lower.  The lift created by the backspin interacting with the air allows the ball to stay up long enough to reach the target even at the lower angle.  The times of flight are noticeably longer.  This is mostly due to the air drag slowing the ball down, requiring more time to cover the necessary distance.

The total time for the two throws is 2.39 seconds as opposed to 2.75 seconds for the one throw home.  The difference is now 0.36 seconds, much longer than the 0.05 seconds for the imaginary airless throws.  In fact, the time difference is now getting close to the time needed to catch, turn, and complete the relay throw.

CutOff

The sketch above shows the flights of all three throws.  The high school physics trajectory without air is in red while, for lack of a better word, the “real” trajectory with air is in blue.  For clarity, the vertical and horizontal scales are different.

You can see the lower launch angles of the real trajectories compared to the high school physics trajectories.  Also, the real trajectory of the long throw home actually goes higher than the no-air trajectory due to the lift created by the backspin.

The rise and the fall of the ball during the throw are no longer symmetrical.  For the high school physics trajectory, the flight peaks at the center of the throw while the real trajectory peaks noticeably later.  In fact, the peak is near the location of the cutoff.  Perhaps this explains the standard practice of making sure the throw from the outfield bounces before reaching home.

In summary, the basic physics of the flight of thrown baseballs once again illustrates that ballplayers are expert experimental physicists (with the exception of the aforementioned Dodgers outfielder).  With almost no knowledge of the theory behind the flight of a spinning baseball, they have discovered that the cutoff play requires very little extra time and, therefore, provides substantial benefits to the defense.

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Comments

  1. Matthew Murphy said...

    Fun analysis. When you add in the increased accuracy by having to make a shorter throw, I think the scales tip decidedly in favor of the cutoff man. However, this also assumes that the outfielder and cutoff man have similar throwing abilities, which may not always be the case.

  2. jake said...

    Nice analysis. In addition, the cutoff system delivers a huge benefit in accuracy (which you could measure in terms of degrees of error in angle translating into feet of distance over various lengths of throw), and allows the OF to deliver a maximum-effort throw rather than balancing velocity and delivery time with a consistent/repeatable/accurate delivery.

  3. Nick said...

    You mention at the end that it is standard practice to let the ball bounce before reaching home. Since you presumably have the code, it would be interesting to search for the optimal distance to bounce the ball. Then you position your cutoff so his chest passes through that path so he can redirect the throw as needed.

  4. bob said...

    If the outfielder throws on the run, the launch speed can be well above 90 mph, even up to 110 mph with a strong arm and fast running speed. Is that enough to make a difference if you assume the higher speed?

    • Steven said...

      Do you have any source that lists 110 mph throws? Players aren’t really able to use the extra momentum from a running start in their throws and often, long toss and off-the-mound velocities are similar for pitchers. In other words, Aroldis Chapman can’t throw 110 on a running start long toss, and he might be the fastest pitcher of all-time. Generally, a little over 100 is considered the highest possible echelon of outfield throws.

    • Steven said...

      For example, Puig’s famous throw to third base last year registered at 95 mph. There are definitely outfielders who can hit 100 or maybe even a couple ticks above, but not 110.

  5. bob said...

    My reference is “The Physics of Baseball” by Robert Adair. He is only guessing when he says that 110 mph is the possible top velocity for an outfielder throwing on the run, and then only for a few players. It would be nice to have actual data, of course, to confirm his guess.

    For a throw of 300 feet, a ball launched at 90 mph needs an elevation of 27 degrees and reaches a height of about 70 feet, while a ball thrown at 110 mph only needs 13 degrees of elevation and reaches a height of 30 feet, again according to Adair. The faster throw arrives about one second sooner.

  6. said...

    This was very neat, I’m actually surprised a relay is even close to a throw-through. In reality, if the throw goes home it’s not cut. Cut-offs are there to prevent the hitter from advancing to second and collect errant throws, not help the ball home. I’d be interested in an analysis of an actual relay situation, say, a ball hit to the right centre gap. In that case the optimal distance for the relay man becomes and inexact science, for sure.

  7. MikeS said...

    In high school, where we did not all have Jeff Francoeur arms, we were taught that hitting the relay man and making two throws instead of one was definitely faster. Few people believed the coach so he put two guys in the outfield and had them both throw at once. The guy with the stronger arm threw it all the way home, the weaker arm threw to an infielder for a relay. The two throws beat the one and were more accurate about three fourths of the time. They only lost when one of the two made a really abd throw, which happened less often than the one guy trying to throw through.

    It is likely that elite athletes mitigate some of the advantage to hitting the relay man, but not all of it in all situations.

  8. Chris said...

    Cool analysis! I’ve played baseball and I’ve tried to look at this with a critical eye. I’m wondering if you can take these into account in your analysis. In many cases, the cutoff man is much less than 90 ft from the target (1B or 3B are always backing up near mound (~70 ft), and so is the SS on a throw to third from RF and CF) so how does that affect the times?

    As well, the BEST catch and release from a major league catcher is approximately .63 – .7 seconds (Game film analysis via Epstein), so I think that time would be quite similar for a middle infielder but not as much for a 1B. Have you done any analysis on the catch and release times here?

    In my opinion, the true benefit of “hitting” the cut is more the act of throwing through him to prevent trail runners from advancing than from actually using him to be faster (or even as fast) as a throw through.

  9. Ken said...

    From a catchers perspective, the reason for the bounce on the throw is to reduce the spin on the ball and make it easier to catch. I realize pitchers have spin on the ball, but the ones from longer distances seem to be tougher to handle with the catchers mitt. At the other bases with a fielders glove, you’d prefer it in the air.

  10. bob said...

    Again, according to Robert Adair, the bounced throw to the plate arrives 0.2 seconds sooner if the field that it bounces off is hard, and that time saved is equal to six feet of the runner’s distance. At worst, if the field is soft, the bounced throw gets there at the same time as the throw on the fly so nothing is lost.

  11. Steve Brosemer said...

    As a shortstop with a better arm than everyone else in the infield, I often wondered why baseball tradition requires the 1st/3rd basemen to be the cutoff man at about 90 feet. There is plenty of time for the SS to move into cutoff position and out a little (edge of grass) so as to make a pair of throws that are more balanced. Using the 270 feet mark, say throws of 150′/120′ respectively. A lower trajectory for the outfielder and then a “normal” medium throw for the SS. Plus, the sooner the cutoff man gets the ball the sooner he can do something with it elsewhere on the infield. Let’s see the math on this theory! Also, on the right side, 1st basemen traditionally have the weakest arm on the infield, so most of this would apply to the generally better armed 2nd baseman on the right side.

  12. Wily said...

    Seems like something is off–not much–but enough to affect the calculations. Even I (an ok high school player) could throw the ball 270 feet with a much flatter (cut-off-able) trajectory than what your figure shows. My guess is that the velocities (with a running start) commonly begin higher than 100. Also the height of the throw is about 8 feet and the receiver is about 1 (for a good throw), which will also change things.

    Probably doesn’t matter too much–biggest difference in both high school and mlb will be relative arm strengths so the optimal strategy varies so much in practice that it’s hard to generalize about these things.

  13. Mark Smith said...

    I like this discussion. It seems obvious that as the distance of the ball from the bat decreases (allowing the OF to make a throw with less distance) that the decision to use the relay man becomes less beneficial. Have you made any estimates as to the switchover point between using the relay and making the long toss? I think we all realize that there are many variables that can make this differ, but in general it’s a great analysis.

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