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Matt Lentzner has carved out a (very) small niche in the baseball analysis world by examining the intersection of physics and biomechanics. He has presented at the PITCHf/x conference in each of the last two years and has written articles for The Hardball Times, as well as three previous articles for Baseball Prospectus. When he’s not writing, Matt works on his physics-based baseball simulator, which is so awesome and all-encompassing that it will likely never actually be finished, though it does provide the inspiration for most of his articles and presentations. In real life, he’s an IT Director at a small financial consulting company in the Silicon Valley and also runs a physical training gym in his backyard on the weekends.

In baseball, much is made of the batter’s ability to see the ball. “See the ball, hit the ball” is a common refrain when hitters are asked about their approach at the plate. But what can seem like a brush-off answer is actually quite profound. It is the essence of hitting.

Bringing the bat to the precise point in time and space required to make contact is, as Ted Williams put it, “the hardest thing in sports.” You’ve got a ball moving at 90-plus mph and a bat moving at 70-plus mph, and they have to meet in such a way that the ball is launched neither too high nor too low, and within the 90-degree wedge that makes up the baseball playing field. Not easy. In fact, the difference between a home run and a harmless pop up is mere fractions of an inch on the surface of the bat.

In order for the hitter to make contact, he first has to see the ball. Then he has to use his hand-eye coordination, strength, and timing to get the bat to that point. As impressive as that physical act is, it is still contingent on the sense of sight. Without seeing the ball, all his hand-eye coordination, strength, and timing is for naught. First and foremost, big-league batters are simply people who see fast-moving objects better than we do.

Indulge me a bit while I take a little detour from baseball to illustrate a point.

About 20 years ago, I was a cadet in the Army ROTC. On one of our training days, we had a “branch day,” on which we would get demonstrations of the different branches of the Army, such as armor (tanks), and infantry. One of the stations was for artillery, and as part of the demonstration I got to assist in a fire mission.

Now, I was excited to pull the lanyard to fire off one of those howitzers. I got to do that. One of the other less glamorous tasks was to take the extra gunpowder from each shot to the powder dump. This involved carrying the bags of powder to a pit located a decent distance away from the gun. Nothing fancy.

But one of the artillerymen gave me some advice just as I was walking away. “Watch the gun when it fires. You can see the round.” This sounded crazy, but I stood patiently by, waiting for the gun to fire after I had dumped my powder. He wasn’t kidding.

You could see a black dot fly off. I watched it for several seconds as it got smaller and smaller, until finally it disappeared. It was as if God himself had hit a giant black golf ball down a mile-long fairway.

This illustrates an important point about vision. It’s not how fast something is moving. That artillery shell was going in excess of 300 mph—easily three times faster that the fastest fastball ever thrown. Seeing something is about tracking, and because of where I was standing, the artillery shell stayed in my field of vision, even though I didn’t move my eyes much. Speed makes a difference, but the location of the observer in relation to the path of the object is key.

The human eye uses two methods of tracking moving objects. One is called “smooth pursuit,” and the other is called “saccade.” Smooth pursuit is when the eyeball moves smoothly to keep up with the moving object. The object is always in full view. The eye can smoothly pursue at speeds of 30 degrees per second, which sounds like a lot, but in terms of hitting a baseball, really isn’t.

It takes about 4/10ths of a second for the ball to arrive at the plate after being released by the pitcher. The eye can smoothly pursue a maximum of 12 degrees in that time period. It would take more like 45 degrees of tracking from where the ball is released to where the batter would typically hit it at the front of the plate. So with a quick back-of-the-napkin calculation, we can see that smooth pursuit won’t work for a large portion of the path of the baseball. When the eye needs to track a really fast-moving object, it uses saccades.

A saccade is when the eyeball “jumps” to a new position. Applied to hitting in baseball, this is when the eye lingers just long enough to form an image of the approaching ball and then jumps to a new predicted point along the baseball’s path to form another image, and so on. This strategy allows for tracking speeds in the 100-700 degrees per second range. Now we’re talking.

A baseball moving at 90 mph will require well over 1000 degrees per second of tracking speed as it passes the batter at something like four feet of distance. So he will lose sight of the ball at some point— that’s a given. But saccades get him a lot closer than smooth pursuit does.

According to the book Keep Your Eye on the Ball: Curve Balls, Knuckleballs, and Fallacies of Baseball (Watts, Bahill), even the best baseball players can track the ball only to within 5 to 5.5 feet of the plate. This roughly translates to a 400 degrees per second tracking speed, which is reasonable based on what we know about saccades. I’m more interested in relative differences for the purposes of this article, so we will assume 400 deg/s is a typical major-league hitter (not a typical person, mind you).

There’s an anecdote from the book Moneyball (I’m sure you’ve heard of it) about submarine-throwing relief pitcher Chad Bradford. According to the book, he was very insecure about his ability to pitch in the big leagues. His fastball topped out in the mid- to high-80s. How was he going to get major-league batters out when top level pitchers were throwing in the mid-90s?

There’s another part of the book where he learns from a third party how a right-handed batter who’d seen him had exclaimed, “There’s no way that pitch was 87 mph!” Once he hears this, he is filled with confidence and goes on to have a solid major-league career. That was the gist if it, anyway. Statistically Chad pitched over 500 innings in relief with a career ERA of 3.26—not too shabby.

Getting beyond the feel-good story of a down-home southern boy who beats the odds with his unorthodox pitching style, just what the heck was going on? How is it that a batter would perceive his pitch to be moving much faster than its measureable speed?

There’s more mystery to Bradford’s pitching performance. Bradford was nearly unhittable when facing right-handed hitters and almost worthless against lefties. Over the course of his career, his platoon splits were drastic: .243/.284/.304 versus righties and .302/.408/.435 versus lefties. Fortunately for him, he faced many more righties than lefties in his career.

The “book” on submarine pitchers is that they are very hard on same-handed batters, but that against opposite-handed batters their pitches seem to “float.” This had baffled me for a long time, so I finally decided to sharpen my pencil and figure it out with my favorite mathematical tools—vector math and trigonometry. 

I don’t want to tax your brain with a discussion of sine, cosine, and tangent. If you want to get an intuitive feel for what’s going on, think of it like this: the maximum angular velocity—and therefore, the hardest point to observe—occurs when the ball is travelling perpendicular to the observer. In other words, it’s the point where the ball passes the observer, changing from approaching to departing. The farther away the object is from this point, the easier it is to see.

Exaggerated view of pitch approach angle. Notice how the
ball passes in front of the batter at a much later point when
thrown from a left-handed release point.

If we exaggerate the situation, then the effect becomes more evident. If you were a right-handed batter and the pitcher were pitching from third base, then the ball would pass in front of you before it reached the plate. If the pitcher were pitching from first base, the ball wouldn’t pass in front of you until after it had passed over the plate. The pitch from first base would be much easier to see at the typical contact point. Note that this is reversed for a left-handed batter.

The actual situation is much more subtle, of course. The difference is a matter of a few degrees. Is it significant? Let’s explore.

First, let’s establish a baseline. On a pitch traveling 132 feet per second (~90 mph) right down the center line of the diamond, the batter will lose sight of the ball 5.41 ft from the front of the plate if we limit his tracking speed to 400 deg/s. What happens if we assume a two-foot deviation from the center line for a right-handed pitcher? The ball approaches at an angle of about two degrees if we assume the target is still the center of the plate. Now he loses sight at 5.56 ft. With a left-handed pitcher, we see a better result of 5.26 ft. This gives us a difference of .30 feet, or about 3.5 inches. Doesn’t seem like much, but we don’t have anything to compare it to.

So what happens when we change the speed? Well, as it turns out, a one-mph difference in pitch speed makes a difference of about 0.5 inches in tracking distance. Now 3.5 inches seems much more significant. We’re talking about an equivalent difference of seven mph.

I don’t want to overstate my case. Pitch speed helps in more ways than just unseeableness (sorry about inventing words, but I’ll be using that one again). Pitch speed gives the batter less time to react, as well. So it’s not exactly an apples-to-apples comparison when I say it’s equivalent to seven mph in value.

This leads us into another detour. There are actually two kinds of “seeing” going on by the batter. One is the batter trying to identify the pitch, and the other is what we’ve been talking about—tracking it to the vicinity of the plate in order to hit it. The decision to swing has to be made within the first third or so of the ball’s path—long before tracking speed becomes an issue. So there are really two separate phases. “See the ball, hit the ball” could be written more accurately as “Identify the pitch, track the pitch, hit the pitch.”

Of course, if the batter misidentifies the pitch, it could screw up his brain’s ability to predict its path and therefore move the saccades accurately. I’ve looked into this, and I couldn’t find any difference in a batter’s ability to identify pitches from opposite and same-handed pitchers. This makes intuitive sense. At the distance where the ball is released, the difference in angles and tracking speeds is negligible. It only matters when the ball is close. Once it comes time to make contact with a pitch—that’s when you see the platoon effect manifest itself. I will explore this further in a future article.

So there are things that a pitcher does to make the ball harder to identify and other things that he does to make it harder to hit. Taking a longer stride will shorten the distance that the ball has to travel, giving the batter less time to identify the pitch. Hiding the ball well has the same effect. Throwing the ball submarine style doesn’t necessarily add to the deception, but it certainly makes the ball harder to track for a same-handed batter.

Speaking of submarine style, let’s get back to Chad Bradford. I estimate that Bradford’s release point was roughly four feet from the center line. Let’s set his pitch speed at 128 fps (~87 mph) and see what we get. My calculation is 5.59 ft for a right-hander and 4.98 ft versus a lefty. That’s a difference of .61 feet or more than seven inches—equivalent to 14 mph! If you can imagine a pitcher who threw 94 mph to right-handers and 80 mph to left-handers, that’s a rough estimate of how hard it was to make contact with pitches that Bradford threw from each side of the plate.

       Note the offset of Bradford's delivery. (Sony Pictures)

You might be wondering how this works in the vertical plane; it’s basically the same. Each degree of offset is equivalent to about 3.5 mph of pitch speed. A typical fastball descends at seven degrees. If you have a really tall pitcher who can release the ball a foot higher than average, then that same pitch will be dropping at eight degrees when it reaches the plate, for an effective speed of 3.5 mph more for the purposes of making contact—just another reason why tall pitchers are so popular. You will also occasionally hear about how short pitchers tend to throw on flat planes. This is why scouts are suspicious of pitchers who don’t have height on their side.

The same effect contributes to why curves are so hard to hit, even though they move so slowly. A typical pitcher with a 90-mph fastball will get his curve over at about 77 mph. But if that pitch is hooking toward the ground at an 11-degree angle, it gains about 14 mph in “unseeableness,” making it slightly harder to track than the fastball. Add in the movement, and the batter is really in a pickle.

My overall goal with the article is to get some relative numbers attached to how batters see the ball and how pitchers try to make it more difficult for them to do so. Presenting things in terms of mph, something we have a feel for, puts the angles involved in some sort of context. Hopefully you can now appreciate the contest between the pitcher and batter just a little bit better than you did before. 

Thank you for reading

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Thanks; very nice article explaining the real-life rationale behind the platoon splits data dichotomy.
You state "a one-mph difference in pitch speed makes a difference of about 0.5 inches in tracking distance". By tracking distance do you mean exactly that, or the actual difference in distance covered from release point to home plate by a pitch. IE a 90 MPH FB will be 1/2 inch behind a 91 MPH FB when the latter reaches home plate? This is great stuff BTW.
No, he's saying that you stop being able to track the slower pitch .5 inch closer to you than the faster pitch.

Over half a second(roughly the amount of time it takes a pitch to reach home plate) the 91 mph FB would be about 8 inches in front of the 90 mph one I think.
Great article. As I'm watching the CWS and UCLA's submarine pitcher David Berg I can't help but think that an over-reliance on standardized pitching instruction at the youth level is keeping some very interesting and useful pitchers from developing a sidearm or submarine delivery.

Read this article for more information on Berg's late developing delivery:

Not sure if linking is allowed here.
I think that an unconventional delivery certainly limits a pitcher's potential. Large platoon splits make it pretty hard to succeed as a starter since it is easy to stack a lineup against them.

It certainly works as a situational relief pitcher, but then those guys are pretty limited in value. I can see even more clearly now why coaches would steer pitchers away from it.

The flip side is that it could be a last best chance for a fringy pitcher to make it. Brad Ziegler comes to mind. Kinda like becoming a knuckleballer. Should certainly be in that "things to try" category.

If your potential is not making it at all... then bombs away. Seems healthier than steroids. I wouldn't mind a 12 year career in MLB as a situational RP. Lighting shoes on fire and hamming it up for the Kiss Cam while gaining eligibility for a MLB pension.
Matt...excellent article, as usual.

I get the essence of what you are saying. However, I am having some problems with the numbers. You say that a 1 mph difference in pitch speed amounts to a 1/2 in difference in tracking distance. Suppose we have a 90 mph pitch (average speed) traveling over, say, 50 ft., so that the flight time is 0.379 sec. If the pitch speed is increased to 91 mph, the pitch will cover 50.6 ft in the same amount of time (0.379 sec). The difference is 0.6 ft or 7", not 1/2". A difference of 3.5" is only a 1/2 mph difference in equivalent pitch speed. Perhaps I am misunderstanding what you are saying, in which case I might be missing the whole point of the paper. Help me out!
Sorry...insert "inch" after the 1/2 on the second line of my comment.
OK, maybe I am seeing what you are saying. You are saying that the tracking distance is the last 5-6 ft or so, in which case I agree with your numbers. However, I now have to think more carefully about your basic premise that the batter's perception of the pitch speed comes from the last 5 ft or so, when the angular velocity is largest.

When I say that 1 mph = 0.5 inches I mean that with two otherwise identical pitches thrown 1 mph different in speed, the faster pitch will hit my arbitrary limit of 400 deg/s 0.5 inches further away than the slower one.

All this stuff happens in a small window about 5-6 feet away from the batter so fractions of inches are still significant.

I am fond of saying that the baseball mind knows nothing of physics. The patterns built in the brain to identify and track pitches are based on timing, perception of spin, and tracking speed. Because of the foreshortened view that batters have we get all sorts on non-physics effects such as balls that hop, "roll off the table", slide, etc.

It's interesting that batters describe pitchers who throw hard as throwing "aspirins" - the ball looks smaller. Makes sense if you realize that they lose sight of it farther away.
Great article.
" Hopefully you can now appreciate the contest between the pitcher and batter just a little bit better than you did before."
Yes I can. Thank you for making something so complex so simple.
Lovely article, thank you. My favorite of your guestuses.
Thanks everyone for the kind comments.
'It's a round ball and a round bat but you've got to hit it square.'--Ted Williams