Why outfielders should throw to the cutoff man, part II

Last sunday, I highlighted a neat article by David Kagan at the Hardball Times that demonstrated the importance of understanding ballistic flight paths when explaining why outfielder should hit the cutoff man.  In my post, I also noted that accuracy on shorter throws by the infielder is also likely important, a point that +Joel Luckhaupt expanded upon when noting that the outfielder's throw is also more accurate when traveling shorter distance.

I thought I'd do some back-of-the-envelope calculations (seriously, I found a spare envelope and sketched this out) to illustrate the point.  Let's assume that any given fielder has a certain throwing accuracy, which we can measure in degrees.  To visualize this, put your nose up to the circle at the center of this protractor:

Our target is at 90 degrees.  If a player's accuracy is + 2 degrees, that would mean that, under stressful, rushed conditions, that player would be able to routinely make a baseball travel within a "window" that is between 88 and 92 degrees.  It's a very small window, but Major League Baseball players are pretty amazing people.

The longer the distance traveled, however, the more a minor angular error in the trajectory he fires a baseball  will result in a ball sailing away from his target.  If we take an outfielder who is 270 feet from home plate, lobbing a baseball at a runner heading home, small angular errors can result in the ball missing the plate by quite a bit:

At 270 Feet
Angular Error (Plus/minus Degrees)	Misses Home Plate By (Feet)
0	0.0
0.5	2.4
1	4.7
1.5	7.1
2	9.4
2.5	11.8
3	14.2

This is where the cutoff man comes into play.  Let's assume (as David did in his article) that the cutoff man is 90 feet from home plate, rather than 270 feet.  Let's further assume that the average MLB fielder's throwing accuracy, under pressure, is +2 degrees.  This is a guess, but it seems sort of reasonable.  Here's what happens (graphic is showing a blimp's-eye view of the field):

The infielder and outfielder have the same throwing accuracy in my diagram: both are +2 degrees.  But the infielder is so much closer than his throws don't have the opportunity to deviate from their target as much.  With some simple trig, we can estimate that the infielder's throws all come in within 3 feet of the plate (roughly the length of a player's arm + glove), whereas the outfielder's throws might come in up to nine feet from the center of home plate.  In other words, the outfielder's throws will be three times as wild as the infielder's, simply because he is three times as far away.

Therefore, not only is hitting the cutoff man allowing a more direct route (lower launch angles negate the time it takes the infielder to catch, transfer, and throw again), and not only does doing so help prevent the hitter from advancing to second base on a throw home, but it also means that the ball will usually arrive in a better location for the catcher to make a play on the runner than if it was thrown all the way from the outfield.

Why outfielders should throw to the cutoff man

A Cardinal Sin of outfielding is to throw home directly, rather than bypassing the cutoff man.  I've always wondered, however, if there are times when that really is the optimal decision for an outfielder.  Allowing the infielder to catch, grab, and then throw the ball must take time, and that's all time that the ball could be advancing toward home plate.  I figured air resistance might be negating some of that gain, but I wasn't sure that it would be enough to matter.

Therefore, I really liked David Kagan's article at Hardball Times last week on the physics of hitting the cutoff man.  David's piece shows that, in fact, one of the major advantages of hitting the cutoff man is that it is a straighter path to get to home plate.  When you throw directly home, you have to use a high launch angle because you have to overcome gravity every step along the way.  Throwing to the cutoff man is a more direct route, and the infielder then has a nice, short throw, again with very little need to elevated launch angle. His diagram shows this best:

Once one also includes factors like air resistance that leads to decreasing velocity during flight (blue line), the total flight time of a ball thrown directly home from 270 feet was 2.75 seconds, compared to just 2.39 seconds.

Now, when hitting the cutoff man, the infielder still needs to catch, transfer, and throw the ball, which must take another half-second to a second (this would be a fun thing to measure...I'll be watching for that in future ballgames).  Therefore, if the infielder is major league baseball-level efficient, we're looking at nearly equal travel times.

Consequently, the benefits of hitting the cutoff man seem to be as follows:

1. Nearly equal time to plate, because the gains of bypassing the infielder are mostly offset by higher launch angle, and thus distance traveled, that are required to get the ball to home plate from the outfield.
2. Improved accuracy by infielder on throw, because slight errors in trajectory angle make less of a difference when you're closer.  I'm guessing this is a really big deal.
3. Ability to prevent the batter from advancing to second base.

Postcript: I'd like to formally apologize to my infielders in 8th grade little league when I bypassed the cutoff man and tried to throw out a batter running to third base.  I didn't succeed, but we might have if I'd thrown to the cutoff man!

Curveballs: illusions and reality

A few years ago, I got to teach a fun freshman seminar course that I called The Science of Baseball.  We covered baseball from a number of scientific approaches: physics, biology (PED's!), psychology, economics, etc.  It was a blast...someday I need to do that again.  I still have the reading list, though it's a bit outdated.

In any case, Michael Maffie had a nice piece today at Redleg Nation highlighting something that is right up that course's alley.  He highlights a PLOS One study that noted an optical illusion that could affect batters' ability to track curveballs.  In the comments, there was also some discussion about why breaking pitches curve.

I wrote this in response, built largely on what I gleaned from Robert Adair's The Physics of Baseball.

Cool stuff. I remember seeing that PLOS One study when it was published, but I’d completely forgotten about it. :)

A few thoughts:
* I think sliders are very interesting in light of this. Unlike the fastball and curve, which will flash as the ball tumbles through space, slider spins are such that the stitches do not “tumble.” Rather, batters see a white “dot” created by the spin as the slider spins on its axis. Therefore, maybe this is yet another difference between what a batter perceives when a pitcher throws a slider vs. other pitches? 

* I have a very basic understanding of curveball (and fastball) physics. But this is the basic breakdown, I think, based on what I read in Adair’s Physics of Baseball. Hopefully it’s correct. 

- The stitches cause the surface of the ball to be “rough.” Rough surfaces create the opportunity for the ball to collect a layer of air around it as it travels at high velocities. This protective layer of air helps to allow for much lower drag than a smooth ball would experience. that effect bottoms out at ~80 mph and then increases again as you increase in velocity. At velocities relevant to baseball, therefore, higher velocity = higher drag = more force applied to baseball. 

- The fact that a curve ball is spinning results in different effective velocities on the top and bottom of the ball. The top part of the ball in a curveball is spinning “into” the wind, and thus has higher velocity, and thus more drag. The bottom part of the baseball, spinning “away” from the wind, experiences less drag. This difference in drag results in more force (called the magnus force) on the top compared to the bottom of the baseball. Therefore, curveballs drop at a (slightly) faster rate than expected by gravity. 

- Fastballs work the opposite way. A four-seam fastball rotates such that the the seams on the bottom of the baseball are rotating into the “wind,” such that the bottom of the baseball experiences greater magnus force than the top of the baseball. As a result, four-seam fastballs (at least) fall less than would be expected by gravity. They still definitely fall (there is no true “rising” fastball), but since we “expect” a normal ballistic path (more or less), they look like they’re defying gravity. 

You can see this deflection on pitchf/x graphs. Brooks Baseball isn’t working for me right now(?!), but if you go here....
Look at the Vertical vs. Horizontal movement graph. Fastballs show up as having “positive” vertical movement, whereas curveballs have “negative” vertical movement. Zero, in that case, would be no deflection from what you’d expect by gravity alone.
Fun stuff!!
-j