The Physics of Throwing a Ball Out of the Yard

It really was a rough inning. When Cleveland’s Trevor Bauer took the mound in the bottom of the fifth against Kansas City on July 29, the Tribe had a 5-3 lead.  On the first pitch of the inning, Alex Gordon singled innocently enough to right field. Then Hunter Dozier hit a high fly ball to center that Oscar Mercado lost in the sun. The ball caromed over the fence for a ground-rule double.

Next, Jorge Soler walked on a 3-2 pitch, loading the bases. Ryan O’Hearn followed with a full-swing dribbler toward the mound.  Bauer tried to use his glove to scoop and flip the ball to the catcher, but he missed the ball altogether, allowing the first run to score. Cheslor Cuthbert hit a weak dinker just past the diving second baseman to tie the game.

Bubba Starling, arguably the best name in baseball, struck out for the first out of the inning. Nicky Lopez slapped the ball up the middle, just past Bauer’s glove, to give the Royals a 7-5 lead.  Just as Bauer was given a new ball to face the next batter, he turned and saw his manager, Terry Francona, heading up the steps with a hook.

All the frustrations of the inning — and perhaps all the stress of the trade rumors swirling as the deadline neared — erupted as Bauer whipped around and threw the ball over the center-field fence. I suspect Trevor realized immediately that he had gone over the edge, because his first words to Francona appeared to be an apology. His manager would have none of it, as he pointed emphatically toward the showers. Take a look.

It is hard to tell from the video just how far the ball actually went. The announcer said he “threw it against the board,” by which I assume he meant the bottom of the batter’s eye.  The center-field fence at Kauffman Stadium is 410 feet. Bauer was maybe 15 feet behind the pitcher’s rubber, while the lower part of the batter’s eye is roughly 10 feet further. Since we don’t know how high up it hit, we can’t get a very exact distance for the throw, but it is probably safe to say the ball went…let’s see…410 minus 60.5 minus about 15 plus roughly 10 plus a bit more…let’s say 350 feet. That’s a homer in many spots in many parks.

I guess it’s not a surprise that a strong-armed player can throw a ball that far. Though it is rarely advisable, an outfielder can throw a ball from deep in the outfield to home plate on the fly.  On the other hand…

At the 1988 MLB All-Star Game, I saw Darryl Strawberry waving to some fans on the top deck at Oakland behind the foul pole. He was standing in the first base coach’s box and fired a strike to the waiting glove of one of those lucky fans. That toss was well beyond 350 feet if it would have reached the ground unobstructed.

Enough frivolity. It’s time for some physics. Let’s begin by estimating the exit velocity (EVO) and launch angle (LA) for a ball with 2000 rpm of backspin that travels 350 feet. There is no unique answer. That is, there are many combinations of EVO and LA that result in a distance of 350 feet. Let’s use Bauer’s average fastball of 94.5 mph for the EVO, which gives an LA of 24.4˚.

Now let’s examine the differences between Bauer’s 350-foot throw and turning around one of his fastballs for a 350-foot hit. Okay, other than the obvious fact that players have a lot more control over the exit velocity and launch angle of a throw than they do when they are trying to create a collision between a thin stick and a small ball moving at over 90 mph. One huge difference is that, during a throw, the ball starts at rest and is sped up to the exit velocity, while a bat must “turn around” a 90-mph pitch and send it off in the opposite direction at roughly the same speed.  That is, the bat must first slow the ball down, then speed it back up in the opposite direction.

The point is, the bat must exert enough force to slow the ball down and then to speed it up again. The player throwing the ball only has to provide the force necessary to speed it up. This is a second reason why it is easier to throw the ball over the fence than hit it there.

Another difference is the time it takes to make a throw as opposed to the time it takes for the bat to turn the ball around. It took Bauer about three frames of one 30-second video to speed up the ball. That’s about 0.10 seconds. Meanwhile, a well-hit ball is only in contact with the bat for about 0.0007 seconds – less than one-thousandth of a single second. The throw requires about 150 times as long to change the speed of the ball as a bat does.

This time difference also has implications for the force required to hurl a ball over the wall. We’ll have to depend upon Sir Isaac Newton to help us calculate the forces for the two cases. Newton’s Second Law says the force on an object can be found by multiplying the mass of the object by the rate at which it changes its speed. Mathematically:

where F is the force, m is the mass of the ball, ∆v is the change in velocity of the ball, and ∆t is the time it takes to make the change.

According to the law, the force due to the throw will need to be smaller than the force from the bat because the throw changes the velocity half as much, and the throw takes way more time to speed up the ball.

Let’s look at the numbers. The weight of the ball is around 5 1/8 ounces (0.32 pounds). For the thrown ball, the change in velocity is 93.5 mph (137 feet/second), and the time is about 0.10 seconds. Beware, if you just plug those numbers in, you won’t get a force in pounds due to the vagaries of English units. Let’s not go into that here. I’ll just tell you the answer is about 13 pounds of force.

For a batted ball, the change in velocity is twice as big for the reasons explained earlier, and the time is about 0.0007 seconds. This means the force the bat exerts on the ball is a whopping 4,000 pounds! You probably didn’t notice, but if you take the 13 pounds for the throw, double it because the bat has to slow down and speed up the ball, then multiply by the 150 times longer the throw takes, you will also get about 4,000 pounds.

These numbers are extremely interesting. It might seem like anyone could generate the measly 13 pounds of force needed to pitch a baseball at professional speeds. The reason it requires huge strong men to do so is related to physiology. The safest thing a physicist can say is that in order to transfer 13 pounds of force to the little bitty ball, you need to thrust a 200-pound body toward home plate. The 4,000 pounds exerted by the bat could lift half an Asian elephant or an average-sized white rhino if the force could be sustained long enough to do so. Incredible!

If nothing else, you’ve here seen the musings of the strange mind of a physicist. While every other baseball fan was thinking about the effect of Trevor Bauer’s behavior on his chances of being dealt before the trade deadline, I was off in another world.