*“All science is either physics or stamp collecting.”*

—Ernest Rutherford (1871 – 1937)

In last week’s column, we started to explore the PITCHf/x data captured by MLBAM’s Gameday application by looking at some basic velocity and strike-zone data. Before moving on to this week’s topic, I’d first like to thank the many readers who offered encouragement and ideas for future analysis. Rest assured they’ve made their way into the ever-expanding queue. This week, spurred on in part by some of your questions, we want to revisit deceleration as pitches make their way to the plate and pitcher fatigue.

**Slow Ride**

Last week I noted that the average pitch (considering the 40,000 or so that had been recorded by Gameday in 2007) starts out at 87.6 mph and ends at 78.8 mph, while the average decrease in velocity is 8.8 mph. When averaging the percentage decrease, the average pitch ends up losing 10 percent of its velocity on the way to the plate. While the average pitch indeed loses 10 percent of its velocity, several readers wondered whether factors such as the release speed and amount of spin on the ball might impact the deceleration of the ball as it speeds towards the batter.

On this subject the logical place to start is Robert K. Adair’s *The Physics of Baseball*. In the second chapter, Adair discusses the effect of air resistance on a baseball. He notes that one of interesting things about baseball, and one of its subtleties, is that the game is largely played with ball velocities ranging between 50 and 120 miles per hour where the air flow around the baseball transitions from “definitely smooth to definitely turbulent.”

Coupled with the raised seams which makes the ball slightly “rough”, the result is that pitchers can make the ball curve, causing no end of trouble for hitters, but also allowing hitters to hit the ball harder and send it farther than would otherwise be possible with a smooth ball; a regular baseball hit 400 feet would only travel about 300 feet if it were smooth. He then goes on to describe the force on a moving baseball as proportional to the cross-sectional area of the ball, proportional to the square of the velocity of the ball, proportional to the density of the air, and proportional to a number called the *drag coefficient* which varies with the velocity of the ball.

By assuming the “density of the air does not vary much for the conditions under which baseball is played,” he produces a graph that plots the variation in the drag coefficient against velocity, along with a second figure that shows the retarding force placed on a baseball at various velocities. The curve that tracks the drag shown in this second figure marks a line with a positive slope that increases in an approximately uniform way between 75 and 100 miles per hour. Taken together, what these graphs indicate is that the faster the ball is thrown, the more drag will be created. Therefore, a pitch thrown at a faster speed should be more affected by air resistance and lose a greater percentage of its velocity after it leaves the pitcher’s hand.

Just how much this effect is in play can be graphed using the Gameday data, since both start and end speed are captured. The following graph shows the percentage difference in end speed for pitches released at various velocities from all thirteen parks for which at least some data exists.

Not all pitches are included, since an attempt was made to choose only straight pitches (pitches with little spin) because of the fact that, as Adair notes, the drag on the ball will differ depending on the axis and velocity of the ball’s rotation. To choose pitches with similar rotational characteristics (fastballs or straight changeups) we can filter by the break length value that PITCHf/x captures (and was introduced into the Gameday application this season). The value is highlighted by the yellow circle on the Gameday display, as in this picture:

This value is defined as the measurement of the greatest distance between the trajectory of the pitch at any point between the release point and the front of home plate, and the straight line path from the release point and the front of home plate. (I assume within a 360 degree radius of the straight line.) Consequently, it not only captures “break” in the traditional sense but also the effects of gravity, so we find that pitches thrown softly such as intentional balls (and the occasional eephus pitch) also have large break values. To account for this fact, I included only pitches with a break length value of less than five and half inches, which typically equates to release velocities in excess of 90 miles per hour, and therefore should include a large percentage of fastballs, as well as a few straight changeups.

As can be seen from the graph, a pitch thrown at a release velocity that ranges from 86 to 100 miles per hour on average loses 10.7 percent of its velocity. However, the amount varies from just over 10 percent in the 86-91 mph range while increasing to 12 percent at 100 mph, verifying Adair’s description of air resistance and the fact that the drag increases with the square of the velocity. In a real sense, then, there is a law of diminishing returns in effect for pitchers, who like everyone else are bounded by physics. As pitchers become better conditioned and are able to release the ball at ever-higher velocities, that gain is increasingly offset by the effects of air resistance.

Last week I also produced a table that showed the average deceleration across all pitches at nine of the parks where over 4,000 pitches had been recorded. We can use the same methodology as in the previous graph to show the difference among these parks in this graph.

This graph reveals a fairly large disparity between parks. San Diego is at the high end, which seems to affect velocity by more than 13 percent for pitches released at greater than 91 mph, while Texas is a little over eight percent at the same velocity. For a pitch thrown at 95 mph, that translates to a difference of 4.3 mph, or in parlance of pitching coaches, around three feet on a good fastball. As mentioned last week, the results seem fairly consistent with the idea that the air density at these parks differ (and are affected by altitude, humidity, weather systems, and temperature), and that accounts for the way in which the ball is slowed down. However, there is a small possibility there could be other considerations as well, including how the system is calibrated at the various parks. For example, note that Anaheim and Los Angeles show a difference, when one might think the conditions would be very similar.

A spinning ball, or one with a differing orientation of the stitches with respect to the direction of motion, will induce greater drag, so we could expect that curveballs and sliders would decelerate more than fastballs or straight changeups. To see this effect, we can examine the pFX value that Gameday also reports for each pitch (to the left of the Break in the screen shot above). This value was in use last year, and was reported as the “break” which has now been supplemented by the break length I discussed at the start.

In any case, pFX is the hypotenuse of the right triangle formed by two other values, the horizontal and the vertical break, that the system also captures. These values are useful, since they break the motion of the pitch into two vectors, which have been explored by Joe P. Sheehan in an interesting article examining the curveballs of Rich Hill and Barry Zito, and are defined as the difference in ending location between the pitch as actually thrown and the same pitch with no spin. So pFX is a measure of the difference between the actual pitch thrown over the plate, and the calculated location of a ball thrown by the pitcher in the same way with no spin. This value is therefore an excellent proxy for the amount of spin on the ball.

With this information we can create the graph below which plots the average difference in speed versus the pFX value for all parks for pitches released at seven different start velocity ranges.

The obvious trend is that as the pFX value increases, so too does the loss in velocity, validating Adair’s commentary that increasing the spin on the ball decreases its velocity as it nears the plate. You can also detect that as the release speed increases, the difference in velocity also increases because of the overall drag on the ball.

This all leads to the question of whether there are individual differences between pitchers in the percentage of velocity their pitches retain on the way to the batter. From a general perspective, what we can say is that pitchers who put more spin on the ball and release it with a greater velocity will lose more on a percentage basis. Conversely, pitchers who impart less velocity with less spin will retain more of their velocity. So in that sense we could certainly expect that pitchers would differ in this regard, since their pitch profiles differ. The question I’m not prepared to answer this week is whether pitchers with similar action on the ball differ in terms of percentage of velocity maintained. My guess is that the physics involved set a pretty firm limit, so any differences, if indeed they exist, would be expected to be small.

**Still Fatigued?**

In last week’s column I also briefly discussed average velocity per inning as an example of the patterns that can be mined and the insights gained with this new data. From that analysis, I concluded that pitchers don’t seem to lose much velocity through their first six innings of work, so it’s likely that pitchers are removed for other reasons–loss of command, a pitch count, a platoon advantage, or even the need to use a pinch-hitter.

However, several readers pointed out that there is a more effective way to slice the data to get at the underlying question. In the previous analysis, I simply examined average velocity per inning, when what would be more effective is calculating the percentage decline in velocity on a pitcher-by-pitcher basis as the game moves along. The following graph looks at all those pitchers (286 in all) who have thrown 80 or more pitches in a game, and plots the change in average release velocity for the duration of their start.

In this graph, the blue line represents the average percentage of the first-inning release velocity for all 286 pitchers. Although the slope of the line seems steep, you’ll notice that it runs from 100 percent down to just 98 percent around the eighth inning. Although there is still selection bias in play, as evidenced by the quickly declining number of pitches in the sixth inning and later, this supports the idea that pitchers only lose at most a couple miles per hour as the game wears on.

In support of this conclusion, we can also average the percentage difference in first-inning and last-inning velocity across all 286 pitchers in order to remove the selection bias when looking only at innings. When we do so, we find that when taken out of the game after throwing 80 or more pitches, the average starter retained 98.2 percent of his first-inning velocity in the inning in which he was taken out. On a 90 mph fastball, that equates to about 1.8 miles per hour.