Spinning Out of Control by Alan Nathan October 26, 2016 Every now and then, something occurs in a major-league game that just compels me to stop what I’m doing, switch gears, and go into analysis mode. It happened most recently in the top of the fifth inning of NLCS Game Five when Kris Bryant hit a fly ball to straightaway — but slightly on the left-field side of — center field. Center fielder Joc Pederson ran nearly straight backward initially facing toward right field. Then he suddenly and perhaps inexplicably spun around to face left field while still running toward the fence. At the last minute the ball went just over the reach of his outstretched glove, on the right-field side of center field. The ball bounced on the warning track close to the CF fence, and when the dust had settled, Bryant was on second base with a double. Just to make sure everything is completely clear: Pederson was initially facing the right direction, then he spun around to face the wrong direction, then he spun back at the last second to the original direction, with the ball barely escaping his outstretched reach. Having spun around a complete 360 degrees, he clearly misplayed the ball. This play generated a lot of discussion on my Twitter feed. I won’t try to summarize all of that discussion. Instead, let me pose several related questions that I will try to answer: Was the trajectory of this fly ball particularly unusual? Is the physics leading to this trajectory well understood? Should Joc have known better? Since this article is not a mystery novel, I’m going to give you all the answers ahead of time: No, Yes, and Maybe. Now for the details. According to Statcast, the ball left the bat with an exit speed of 107.1 mph, a vertical launch angle of 20.8 degrees, and a direction of 2.4 degrees to the left of dead center field. The ball landed 382.6 ft from home plate at an directional angle of 4.4 degrees to the right of dead center field and with a hang time of 3.90 seconds. Even more informative is the actual trajectory measured by the Statcast system, which was provided to me by a source who prefers to remain anonymous and which is plotted in Figures 1-3. In all three figures, the small red dots are the actual tracking data, with the black squares indicating the position of the ball at 0.5-second intervals. The line is a fit to the data using my aerodynamic model for the flight of a baseball. Figure 1 shows a side view of the trajectory, where the height is plotted versus distance from home plate. There is nothing particularly unusual about this view of the trajectory: the ball goes up, the ball goes down. The fact that the line goes through the data points is an indication that the model correctly accounts for the features of the trajectory. Figure 2 is a plot from a top view, showing the horizontal motion of the trajectory, along with the diamond and foul lines. Figure 3 also shows the horizontal motion, with the scale greatly expanded so that more detail can be seen. It is clear from these plots that the ball leaves the bat in a direction just a bit to the left-field side of dead center, then starts to break in the opposite direction, toward right field. According to Statcast, the ball started out 2.4 degrees to the left of center field and ended up 4.4 degrees to the right of center field, so that the total amount of horizontal break was 6.8 degrees. Given that Bryant is a right-handed batter, the break to the opposite field is commonly called a “slice.” Had the ball broke toward the pull field, it would be called a “hook.” This is exactly the same terminology used in golf. Once again, the fact the the line mostly goes through the data is an indication that the trajectory is being properly modeled. According to the model, the ball leaves the bat with approximately an equal mixture of backspin and sidespin, the latter being the cause of the slicing motion that seem to cause Joc so much confusion. So now that we know the details of the trajectory, we come to the first question I posed: was the trajectory of this fly ball particularly unusual? To answer that question, let’s take a look at more Statcast data, as shown in Figure 4. Using data from the 2015 season, I have plotted the final direction versus the initial direction for balls hit with similar launch parameters to Bryant’s hit, namely (a) an exit speed in the range 100-110 mph and (b) launch angle in the range 15-25 degrees. Each circle represents a ball in play. Balls hit by right-handed batters are plotted as shown; for left-handed hitters, I simply changed the sign of both initial and final directions. Thus balls hit in a negative direction are “pulls,” those in a positive direction are “pushes”. The red line corresponds to final direction equal to initial direction. Points above the line correspond to slices, those below the line correspond to hooks. Now take a look at balls hit in the vicinity of straightaway center field (i.e. initial direction close to 0 degrees). Virtually every point in the plot falls above the line, meaning those balls are sliced: they break to the opposite field by an average of about 6-7 degrees, just about exactly the same as the Bryant hit. So the answer to the question is pretty obvious: there is absolutely nothing unusual about this trajectory. Indeed, hard-hit balls in the air to straightaway center field almost always slice to the opposite field. The “crossover” point is about -10 degrees, for which there is neither a slice nor a hook. Balls hit to the left of -10 degrees tend to hook; those hit to the right tend to slice. Let’s move on to the next question: is the physics leading to this trajectory well understood? If physics-y things do not interest you, please feel free to jump ahead. For those of you still hanging in, let me refer you to the excellent article that my buddy and fellow physicist Dave Kagan wrote a couple of years ago, The Physics of a Foul Down the Line, where all the physics you need to know to understand the trajectory is explained. There are three key points to understand. First, for balls hit to straightaway center field, the axis of the bat is nearly perpendicular to the direction of the pitch; that is, the batter is neither out in front or behind, as shown in Figure 2 of Kagan’s article. Second, for fly balls hit to the outfield, the bat is almost always below the center of the ball, which creates backspin and lift. I refer you to a previous article I wrote, Optimizing the Swing, in which this is all explained; in the language of that article, the attack angle falls below the centerline angle. Third, and this is really the key point, the bat is almost always tilted downward so that the barrel is below the knob. Indeed, if you look at Figure 5, a screen capture from the video, you can see that Bryant’s bat is tilted down at close to a 45-degree angle at the moment of impact. Now take a look at Figure 6, a pitcher’s view of the ball-bat collision by a right-handed batter. The left side shows a level bat oriented perpendicular to the pitch and hitting the lower half of the ball. The arrow shows the direction that the front, or leading, edge of the batted ball is turning due to the spin, which results primarily from the frictional force exerted by the bat on the opposite, or contact, side of the ball. The upward arrow denotes pure backspin. Remembering that the Magnus force on a spinning baseball always points in the direction that the leading edge is turning, pure backspin results in lift on the struck ball. Now compare that to the right side, which shows the bat tilted downward. The spin axis tilts right along with it, so that the leading edge of the ball is rotated as shown, corresponding to a mixture of backspin and sidespin. That is, the leading edge of the ball is pointed up and to the pitcher’s left, leading to both lift (“up”) and slicing motion (“left”). If the bat were tilted downward at a 45-degree angle, it would result in an equal mixture of backspin and sidespin. Incidentally, that is exactly what would account for the trajectory in my aerodynamics model. I come to the conclusion that the physics of this batted ball is very well understood. Now we come to the third question: should Joc have known better? Of course, Joc is not a physicist and he should not be expected to understand why the ball does what it does. However, as an experienced outfielder, he surely ought to know from many observations that fly balls hit to center field almost always slice. Therefore, if a ball is hit directly at you, as this one was, the first move should be to turn toward the slice side. Joc actually did that, but somewhere along the line he changed his mind and subsequently turned the other way. I don’t know what he saw in the trajectory that led him to do that. My verdict: since his first move was correct, I give him a pass and answer the question with a definite “maybe.” Thanks to Tom Tango for calling this play to my attention and to my anonymous benefactor for providing the Statcast data.