The Physics of Baseball in Super Slow Motion

A bat is roughly seven times more mass than a baseball. (via Jull Carter)

A bat is roughly seven times more mass than a baseball. (via Jull Carter)

There are few images as stunningly poetic as baseball in super slow motion. As more and more broadcasts include this type of footage, I thought it might be nice to provide a primer on the physics illuminated by these images. Don’t misunderstand, directors and producers include super slow motion for their visual beauty, not their scientific value. Nonetheless, there is lovely physics a plenty.

Basic video is shot at 30 frames per second. In other words, there is one-thirtieth of a second between individual frames. In this time, a thrown ball travels about four feet. A well-hit ball leaving the bat covers a slightly larger distance, while the bat itself moves slightly less. A runner sprinting toward first advances about one foot.

So much of the detail of the game happens in much shorter times. A typical curveball completes a rotation in less that one-fiftieth of a second. The collision between the bat and the ball can cause the bat to vibrate back and forth in around two hundredths of a second. The ball is only in contact with the bat for a little less than one-thousandth of the second.

Slow motion is made by taking video at higher rates and playing it back at the 30 frames per second required for a broadcast. Typical cameras around the park actually shoot about 240 frames per second. Only one of every eight frames is shown so that we see the action at 30 frames per second and the action appears at normal speed.

If the 240 frames per second were all shown at 30 frames per second, we would see the action in slow motion – actually eight times slower than real time. Are you following the math here? Slow motion is created by capturing images at rates higher than 30 frames per second and then showing them at 30  frames per second.

To see the ball colliding with the bat, you would need to shoot at least one thousand frames per second so each frame happens in one-thousandth of a second. The super slow motion we see in broadcasts is shot at five thousand frames per second, so we should be able to see the spin on a curveball, the vibrations of the bat, and even the ball-bat collision. Now, back to physics.

The Law of Conservation of Momentum

“The total momentum of a system of objects is the same before a collision as it is afterward.”

Since the bat is at rest in the video above, it has no momentum. The ball has momentum toward the catcher. Thus, the system of the ball and bat together has momentum toward the catcher before the collision. According the law, the momentum after the collision must also be toward the catcher.

When the ball bounces off the bat, the ball now has momentum toward the pitcher. To satisfy the Law of Conservation of Momentum, the bat must recoil back toward the catcher to maintain the total momentum of the bat and ball in that direction. You can see that indeed it does.

The home run situation is more complicated because both the bat and ball have momentum before the collision. The momentum of the bat is toward the pitcher, while the momentum of the ball is toward the catcher.

The bat is about seven times more massive than the ball, although the ball is moving about 30 percent faster than the bat. It turns out the momentum of the bat is a bit more than five times that of the ball. Since the bat has more momentum toward the pitcher than the ball has toward the catcher, the total of the bat and ball together is toward the pitcher.

After the collision, the momentum of the ball has changed from being toward the catcher to toward the pitcher. The law requires the total momentum of the bat and ball toward the pitcher to be the same as before the collision. Since the ball now has momentum toward the pitcher, the bat must lose some of its momentum toward the pitcher and therefore slows down.

Newton’s Third Law

“Whenever one object exerts a force on a second object, the second object exerts an equal force back upon the first object.”

You can see the pitcher’s arm bend in an unnatural way as he delivers the pitch. Try as you might, you can’t bend your arm that way by yourself. You need something pushing back hard on your hand to deform your arm that much. In this case it is the ball. We see Newton’s Third Law in action: “Whenever the pitcher’s hand exerts a force on the ball, the ball exerts an equal force back on the pitcher’s hand.”

A Hardball Times Update
Goodbye for now.

The Magnus Force

The Magnus force is caused by the interaction of the spin of the ball on the air around it. The Magnus force explains why objects with topspin tend to fall faster than objects with backspin. The spin on this pitch creates a Magnus force that causes the ball to drop and move in on the right-handed batter. You can read more about the Magnus force right here at THT.

The Compression of the Ball

If you took a basic physics class, you might remember the instructor always avoiding issues associated with the inherent elasticity of objects. The class seemed to deal only with rigid objects.

It turns out physics is perfectly capable of addressing the behavior of non-rigid bodies, but it is really difficult. So instead of going into detail here, just enjoy this amazing video. At the speeds involved in the collision, the ball is highly elastic. It looks like it is made of rubber!

You might think that crushing the ball and having it flex back into the sphere it once was would reduce the amount of energy available for the ball’s flight. You would be correct!

The Law of Conservation of Energy

“Energy can be changed from one type to another, but the total remains unchanged.”

This law requires the energy just before the collision of the ball and bat to be equal to the total energy afterward. Since the smashing of the ball during the collision reduces the energy available to propel the ball, it must show up somewhere. Some of the energy is converted to heat.

The infrared camera used in the video is designed to detect heat. You can see the bare skin of the batter is giving off more heat than is emitted through his clothing because the clothing appears darker. The heat created in the collision manifests as the white spots on both the bat and the ball.

If you remember this play, there was a bit of controversy about where the ball hit the batter’s toe. Look closely at the video and you will have the definitive answer.

Vibrating Bats

Another way energy is lost in the ball-bat collision is through the vibrations of the bat. After all, getting a bat to vibrate like that takes energy – energy that would otherwise be available to drive the ball.

It really stings when you hit the ball off the end of the bat. This video lets you see those vibrations in the bat that cause the feeling known as “a handful of bumble bees.”

The Sweet Spot

When the ball is hit on the “sweet spot” there are few vibrations in the bat. So, no energy is spent on vibration, leaving more available to speed the ball on its journey to the cheap seats.

Breaking Bat

The vibrations in the bat caused by a poorly hit ball actually can be so large that the fibers of the wood can’t take the strain. That’s a broken bat.

In this video, you can see the wave created by the ball hitting the end of the bat travel up to the handle. The barrel of the bat is strong enough to deal with the wave, but the handle is not.

Well, Einstein, that’s today’s physics lesson. A big thanks to the folks that do the amazing work to make the invisible, visible.

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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Rick Sharp
8 years ago

Very cool footage and nice, clear explanations of some relatively complicated physics. Will definitely be checking out your site! Would love to see Napoli’s spring training, broken bat HR in super slow mo. Also would be very interested in the physics behind different pitches, catchers framing and their receiving and throwing to a base.

David Kagan
8 years ago
Reply to  Rick Sharp

There is a nice analysis of the Napoli broken bat homer at Alan Nathan’s site – The Physics of Baseball ( Enjoy!

Buckeye Engineer
8 years ago

Great, succinct article. Should be required reading for high school athletes that don’t think physics is interesting. Here at Ohio State’s College of Engineering, we produce a video series called Athletics Engineered. Our curveball video ( dives into the Magnus Effect and the Bernoulli Equation. Our tennis video ( serves up some sweet spot science. If anyone has other ideas for us to cover, send them my way!

Peter Denton
8 years ago

Physicist checking in here:

Off the bat, a typo: “A bat is roughly seven times more massIVE than a baseball.”

Okay, their discussion of momentum is lousy. Yes, it is true that momentum is ALWAYS conserved (despite what my students may or may not try to claim on tests), but you must keep in mind, in this case, both the momentum of the hitter and the momentum of the earth. The fact that a hitter’s feet are firmly planted is important (they wear cleats for a reason). A person’s arms aren’t rigid, they pass some of the momentum to the body and then the earth. (In fact there are no such things as rigid objects, but that’s a lesson for a class of special relativity.)

Alan Nathan
8 years ago
Reply to  Peter Denton

I’m a bit late jumping in here.

Commenting on conservation of momentum, the bat is essentially a free object on the short time scale of the collision. Therefore momentum conservation (as well as angular momentum conservation) only includes the ball and bat. Lots has been written on this, particularly on my website: The non-rigid nature of the bat and the shortness of the collision time are the key factors here.

Commenting on the OSU videos: Your colleague Mike Lisa from the Physics Department has just published a superb textbook on the physics of sports: Expensive but worth it (my copy was free!)

8 years ago

u suc ass