The Physics of a Bad Hop

Plays like the one in the video below are why DJ LeMahieu is considered one of the best second basemen in baseball. (via Keith Allison)

When we watch a major league infielder collect a routine grounder and fire to first, I’m sure some of us think, “I coulda been a ballplayer.” It is when they have to deal with sharply hit one-hoppers that we feel much more comfortable sitting on the sofa, watching the big screen, rather than risking life and limb.

Check out this play by DJ LeMahieu on a ball that left the bat at 102 mph. He made the play despite not only the velocity of the ball, but it taking a bad hop as well.

I wanted to explore the physics of these bad hops. But before we get into that, perhaps we should understand the physics of a “good hop.” Andrew Dominijanni described good hops in two recent articles at The Hardball Times: The Physics and Timing of the Infield Bounce Throw and The Physics and Timing of the Outfield Bounce Throw. For his analysis, he used the standard equations describing the bounce of spinning ball. I include them here in case you are an “equation person.” If not, ignore them and skip to the next paragraph where I’ll attempt to explain them in English.

The horizontal speeds of the ball before and after the bounce are vx1 and vx2. These values tell us how much the ball slows as it moves toward the fielder due to the hop. The vertical speeds of the ball before and after the bounce are vy1 and vy2. These values tell us how high the ball will pop up after the bounce. Finally, the spin of the ball before and after are ω1 and ω2.

Of course, the speeds and spin of the ball after the bounce depend upon the speeds and spin before. In addition, the result depends upon some numerical factors (R, the radius of the ball, 1 and 0.4), and, most importantly, two key parameters: ey, the coefficient of restitution (COR), and μ, the coefficient of friction (COF).

The COR is related to the bounciness of the ball. It can be measured by firing the ball into a wall while measuring the speed before and after the collision. The ratio of the speed after the collision to the speed before the collision is the value of the COR. For a baseball colliding with a wall made of ash at 60 mph, the COR is supposed to be between 0.514 and 0.578.

The COF is the ratio of the horizontal force on the ball during the hop to the vertical force on the ball during the hop. The horizontal force is usually called “friction,” while the vertical force is called the “normal force.” Normal here doesn’t mean usual or common; instead, it means perpendicular to the ground. The COF is much harder to measure, but it is around 0.4 for a good hop.

If the ball bounces off the infield grass as opposed to the dirt, the hop will be less pronounced because both the COR and COF are smaller for the grass than the dirt. Yet this difference should not be considered a bad hop; the infielders have a sense of this distinction due to decades of experience fielding.

At this point, you might be thinking that there is a simpler explanation for bad hops. After all, where you and I play, the infields are full of stones, tufts of grass, and even the occasional gopher hole. In the big leagues the infields are remarkably well tended, with the dirt groomed every three innings. As a result, the COR and COF are probably reasonably consistent.

So variations in the COR and COF will not likely be the explanation for bad hops. The explanation may be far more subtle. The equations for a good hop are put together using some assumptions that aren’t generally discussed. However, they have implications for a bad hop.

Below is a list of a few of these assumptions that are relevant:

  • The infield surface is flat.
  • The ball is a rigid object and doesn’t compress during the bounce.
  • The infield is also a rigid object and doesn’t compress during the bounce.
  • The infield also doesn’t move horizontally during the bounce.

Let’s consider the effect of each of these assumptions on a well hit ball that bounces.

Each of these assumptions is wrong. For example, those gopher holes near second base destroy the flatness assumption, although texture on major league diamonds is unlikely to be the culprit, as the dirt is so well cared for. Of course, the fielder’s cleats can cut up the field a bit, but those small deviations will likely be smashed flat by the ball.

A Hardball Times Update
Goodbye for now.

Further, the ball does compress to some degree. In fact, during the collision with the bat, the ball can compress to as much as half of its diameter. However, most bad bounces occur on sharply hit one-hoppers where the ball hits the ground at a very small angle. In other words, the ball is moving downward very slowly because it is almost completely moving toward the fielder. So these sorts of collisions won’t cause the ball to compress very much at all.

The field can also compress – the grass more so than the dirt. This is part of the reason the COR for the grass is smaller than the COR for the dirt. Here is where a cleat mark might affect the bounce of the ball. The cleat might loosen the dirt to the point where dirt compresses substantially during the collision. The would cause the ball to stay down. So, we might have an explanation for a smaller than expected bad hop.

Lastly, the top surface of the dirt can slide horizontally. This is true regardless of whether cleats have cut up the dirt or not. Of course, the more cut up the surface, the more dirt there is to slide. Now, imagine a low trajectory line drive hitting the dirt. The ball might skid along the ground and move the loose dirt into a small pile in front of it, as seen in the exaggerated sketch below. This effect might even be enhanced due to the spin on the ball. This pile of dirt now acts as a launching pad to produce a bad hop that goes higher than expected.

Hopefully, these explanations of bad hops here have some relevance to actual bad hops. Since there is no experimental data available to test these ideas, I suppose they are safe for now. However, the most important point here is to remember that equations, while wonderful tools for understanding, are only as useful as the assumptions that go into building them.

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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3 years ago

Is the infield’s surface ever non Newtonian in nature at some velocity or spin rate?

3 years ago
Reply to  David Kagan

I wondering if flow does occur, is resistance to flow linearly related to impact velocity. or if shear forces behave in an odd way.

Jetsy Extrano
3 years ago

Neat ideas! I had always assumed that bad hops come from deviations in effective ground surface normal (i.e. pebbles, clods), but mechanisms like the collision shifting the dirt are interesting.

Any chance somebody with a pitching machine can run some experiments bouncing it and seeing the distribution of bounce? How much do bad hops deviate vertically versus horizontally?

Jetsy Extrano
3 years ago
Reply to  Jetsy Extrano

Do you have the pull that you might be able to get access to Statcast data to publish on actual in-game hop behavior? That would be awesome.

Jetsy Extrano
3 years ago
Reply to  David Kagan

Do you know how far that extends from the ground? Yeah, that might interfere with getting the bounce path for grounders that are fielded off the hop.

Sam Sharpemember
3 years ago

You mention two of the variables in the equation, but what is R?