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|
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|
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.
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.