The Physics of Broken-Bat Homers

Mark Teixeira has popped a broken-bat homer or two in his day. (via Arturo Pardavila III)

Mark Teixeira has popped a broken-bat homer or two in his day. (via Arturo Pardavila III)

They seem to be a little less rare than no-hitters, but there are two differences between broken-bat homers and no-nos.  While you can find many thorough listings of no-hit games, there is no authoritative source for broken-bat homers.

David Schoenfield and Rob Neyer of ESPN.com tried to build a list in 2009 by asking their chat followers to recall as many as possible. They came up with about a dozen names, including one from Mark Teixeira

However, vague memories hardly make a definitive list.

The other difference between a no-hit game and a broken-bat homer is one can be explained by great pitching and good fortune, while the other can be understood through great hitting and physics.

First we need to understand why bats break in the first place.  A power fastball hurls toward the batter at about 95 mph, while a homer leaves the bat around 105 mph.  So, the bat must change the velocity of the ball by roughly 200 mph.  Alan Nathan’s investigations show us that a well-hit ball is in contact with the bat for about half a millisecond.

Isaac Newton was the first to realize how to find the force needed to accomplish this change in velocity, which is really quite remarkable since baseball hadn’t been invented yet!  According to Newton’s Second Law, the force on the ball is the mass of the ball times the change in velocity of the ball divided by the time of the collision.

Since I assume you aren’t interested in watching me do math, I’ll just tell you the answer – about 5800 lbs.!  That’s right…and I double-checked the calculation.  There really is almost three tons of force on the ball.  Wow, that’s about half the weight of a bull elephant!  You can choose which half.

We actually don’t care about the force on the ball.  After all, it is the force on the bat – not the force on the ball – that causes the bat to snap.  Fortunately, Newton explained how to do this with his Third Law.  The force the bat exerts on the ball is equal to the force the ball exerts on the bat.  No math required: there is half a bull elephant exerted on the bat.

No wonder bats break sometimes. In fact, it’s amazing they don’t break on every home run!  When bats break, they don’t usually exert enough force on the ball to hit it out. Typically, the ball comes off the bat as a slow roller to the infield or a blooper into the outfield.

The famous broken-bat hit by Hunter Pence in the 2012 NLCS is noteworthy not just for the fact that it happened in a crucial situation, but the details of the collision were clearly revealed with an ultra-slow-motion camera.

When the ball first hit the bat, the bat only exerted enough force to stop the ball in midflight – maybe 2,000 lbs. – certainly nowhere near the 5,800 lbs. needed to send the ball off toward the outfield.

It wasn’t until the broken end of the bat came around and struck the ball two additional times that the ball began its fateful journey toward the outfield. The additional force – as well as the unusual spin imparted to the ball by the two extra collisions – sent a weak grounder into playoff history. But we’ve drifted off topic.

A Hardball Times Update
Goodbye for now.

How can a bat made of wood exert such a large range of forces? When the ball hits the bat near the narrow part, the bat flexes during the collision. This reduces the force in the same way the flexible padding in your glove keeps the forces smaller than they would be if you caught the ball with your bare hand. The Physics of the Pitcher’s Padded Cap explains this effect in some detail.

The bat flexes a lot less during the collision if the ball hits on the barrel. The resulting force on the ball will be larger. This explains why broken-bat hits off the barrel are generally stronger than broken-bat dribblers off the handle.

The funny thing is, no matter where the ball collides with the bat the, bat breaks on the narrow part of the handle. I guess it makes some sense the bat would break at its weakest part, but how does the effect of a force exerted on the barrel get transferred down the bat? The answer is waves.

When the ball strikes the barrel of the bat away from the sweet spot, a wave is created on the bat. This wave travels down the bat toward the handle. These days, it’s not uncommon to see these waves propagate down the bat during broadcasts that use ultra-slow-motion cameras. Just to see what’s going on, take a look at this footage shot in the laboratory from TimeWarp.

The small wave created in the thick barrel of the bat grows in size as it moves toward the more flexible handle. This is the same thing you see when you’re lying on the beach. The small swells out in the deeper water grow as they approach shallow water near the shore. What a wonderful linguistic coincidence that we say waves “break” as they approach the shore, just as a bat breaks when the wave gets to the handle.

For the ball hit by Pence, the ball directly broke the bat. The bat flexed during the collision, so the collision time was much longer than the collision with the barrel in the TimeWarp video. The longer collision time is consistent with the smaller force exerted on the ball, while the shorter collision times for broken-bat hits off the barrel result in larger forces on the ball.

The other thing to notice in the TimeWarp video is the ball is no longer in contact with the bat by the time the wave gets to the handle and breaks the bat. That’s right­, the ball was headed for the bleachers by the time the bat snapped.

The ball doesn’t even know or really care that it broke the bat. It behaves just as if the bat didn’t break and flies off without a care in the world. However, a ball directly responsible for breaking the bat near the handle slinks off as if it has a guilty conscience.

So, next time you see a broken bat result in a weak dribbler, check out the replay. It will most likely show the batter hit an inside pitch near the handle. On the other hand, if you’re lucky enough to see a broken-bat homer, the replay will show a ball hit further up on the barrel. Oh, and make a note for your new authoritative list of broken-bat homers.


David Kagan is a physics professor at CSU Chico, and the self-proclaimed "Einstein of the National Pastime." Visit his website, Major League Physics, and follow him on Twitter @DrBaseballPhD.
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baseballfan123
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baseballfan123

Cool article. It must be mentioned sometimes bats break because the ball makes contact in the “weak” part of the barrel which is perpendicular against the wood layers instead of parallel with the layers. Usually the branded logo on wooden bats is printed in the “weak” part of the bat, therefore if you hit it a ball near the logo the bat could break

joser
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joser

The ball is always headed for the bleachers (or wherever it is headed) by the time the wave hits the handle. So nothing that happens at the handle has any influence (at that point) on the velocity of the ball. Which is also why something like Frazier’s no-handed HR is possible.
(Alan Nathan did a presentation on this, the video of which is at MLB.com).

David Kagan
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Thanks for adding this information – Alan is wonderful. Here is my slightly more detailed and much more humorous description of Frazier’s no-hands homer:

David Kagan
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Kerry
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Kerry

I saw Bo Jackson break his bat on a pitch, and the ball was caught at the base of the wall in dead center field, 410 feet away in Kaufmann stadium. Longest I’ve ever seen a broken bat hit go. Would have been a HR anywhere else.

Alan Nathan
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Alan Nathan

My discussion of the “grip doesn’t matter” issue (including the Frazier no-hands HR) is here:
http://baseball.physics.illinois.edu/grip.html.

My discussion of a broken bat HR is given here:
http://baseball.physics.illinois.edu/BrokenBatHR.html.

The paper upon which the theoretical analysis is based is given here:
http://baseball.physics.illinois.edu/AJP-Nov2000.pdf

Enjoy reading!

Don
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Don

UMASS-Lowell has a “baseball research department”? That alone has restored my faith in the American educational system!