# The Physics of Hard-Hit Balls

*Editor’s Note: This piece was initially given as a presentation at the marvelous 2016 Saberseminar.*

If you look at the Statcast Leaderboard, Giancarlo Stanton is as dominant on the hitting side as Aroldis Chapman is on the pitch speed chart. Yet, he has not earned a “Stanton Filter” like Chapman has on the pitching side.

Stanton has topped 120 mph for the speed of the ball of his bat. For some reason, MLB calls this the “exit velocity” as opposed to the “batted ball speed” or something more sensible. Although in Stanton’s case his hits act as if they were shot out of a cannon, so I guess “exit velocity” is appropriate.

On April 30, 2016 Stanton hit a mammoth 462-foot homer. According to Statcast, it had an exit velocity of 116.8 mph at a launch angle of 23.5˚. These numbers seem to be valid and within tolerance, matching the values from ESPN Home Run Tracker.

Now that a good portion of 2015 Statcast data is available, we are in a position to look at the physics of all well hit balls – be they homers, singles, or loud outs.

Below is a plot of all balls hit by Stanton in 2015 with an exit velocity over 100 mph. You might recall that this sample is limited because of the frightening bean ball he took to the face the previous year.

You’ll notice the homers form a cluster with launch angles between 15˚ and 40˚ while the speed varies from about 100 mph to 120 mph. There are plenty of well hit balls in the 100 mph to 120 mph range that are not home runs due to launch angles that are too high or too low.

A second key feature of this plot is the range of angles is larger for balls with lower exit velocities while the range is smaller for harder hit balls. Let’s see if these two features are consistent by looking at data from a couple other hitters.

For Bryce Harper we have more data than for Stanton. Again we see the homers cluster although the range of angles and speeds is a bit larger. There are several balls hit harder than the hardest hit homer. The greater spread of angles at lower speeds is also evident, but a bit less compelling.

Mike Trout’s plot yet again shows the clustering of homers with some harder hit balls not leaving the yard and the spread of angles at lower speeds. This time I have sketched, more or less, the envelope that seems to indicate the extremes of the spread of angles with lower speeds. At this point I can’t help myself, I need to plot this data for all 2015 and begin to think about the underlying physics.

### The Clustering of Home Runs

You may have already guessed the reason for clustering of home runs and the fact that many harder hit balls don’t result in a round-tripper. You need to get the right amount of loft on the ball. Too little and it won’t make it to the fence. Too much and it is just a lazy fly ball.

Looking at the plot below of launch angle versus exit speed for just the homers in 2015, you can see the clustering. The black regression line summarizes the fact that the higher the speed, the lower the necessary launch angle. The lower launch angles describe the line drives that “get outta here in a hurry,” while the higher launch angles are the majestic fly balls that that can take up to five seconds to find the cheap seats.

The line drives are in the air only a short time so they can’t travel as far as a fly ball. The high fly has plenty of time as it makes its way to the fence, so it need not be going as fast. An interesting point to consider is that a fly ball homers actually have two reasons why they stay in the air so long. The first is the larger launch angle.

The second reason relates to the details of the ball-bat collisions. Hitting the ball off the top half of the bat creates the large launch angle. Hitting the ball off the top half of the bat also imparts backspin to the ball. This backspin results in additional lift keeping the ball in the air longer.

### The Spread of Angles At Lower Speeds

Here is the launch angle versus exit velocity for all of MLB in 2015. I ran out energy for collecting this huge data set when I got down to 100 mph. As a result, the lower end of the home run cluster isn’t shown. However, the spread of angles with lower speeds is very clear. Let’s go a bit deeper into the physics of this effect.

I assume you have played pool or billiards. If on a given shot your goal is to maximize the outgoing speed of the ball you are trying to hit, you know that a direct hit by the cue ball is the way to go.

On the other hand, suppose your goal is get the ball you’re aiming at to head off at a steep angle to the incoming cue ball. You know that the ball will head off at a much slower rate than the direct hit. In summary, a direct hit by the cue ball results in a small angle and high speed, while the less direct hits result in higher angles by lower speeds.

If you think of the bat as the cue ball and the baseball as the ball you’re aiming at, it isn’t too surprising that the result is the same. The gory details of the ball-bat collision can be found in Alan Nathan’s paper.

To calculate the launch angle versus the exit velocity, I used a simpler model of the physics of the ball-bat collision that depends upon the following quantities:

- The weight of the ball
- The weight of the bat
- The incoming pitch speed
- The speed of the center-of-mass of the bat
- The faction of mechanical energy lost in the collision

The results give curves shaped like the one I just sketched in Trout’s plot. I fiddled around with the four quantities to get a curve that fit the complete set of 2015 data and it is shown in the plot.

The quantities that produced the curve shown are:

Quantity | Value |

Weight of the ball | 5.25 oz |

Weight of the bat | 36 oz |

Pitch speed | 90 mph |

Bat speed | 70 mph |

Energy fraction | 0.58 |

All these values are pretty reasonable – that is to say, consistent with values found by other methods. So physics explains not only the spread of angles at lower speed, but also has a small bonus feature at no extra cost. Notice the curve is not at the peak exit velocity at a launch angle of zero. The value I used in the curve is 9˚.

I suspect this is due to the fact that despite years of coaches telling players to “take a level swing,” they have a small uppercut instead. So maybe the moral of this story is “listen to your physics professor, not your coach.” Somehow, I’ve got a bad feeling about that advice…

Enjoyable and very informative! I especially liked your final comment as to whom to listen. My background is medical and I can remember coaches refusing to allow players to hydrate…

There’s a whole other overlay, too: environmental context. More homers in July than any other month; the air, breeze, and humidity combine to give the batter a better shot. Except indoors, one assumes.

I wanna see a metric of which part of the bat makes contact, as a z axis, on this. And what is the variation in bat speed for various types of hits? (Yes, I am a troublemaker.)

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