About five years ago, I was gifted a wonderful book with the title of What If? In it, Randall Munroe tries to tackle random hypotheticals with simple math and humor. I think you could point at this book as one of the things that fueled a fascination with numbers that continues to this day. While I will probably never be as smart as Munroe, who worked as an engineer for NASA or match his simple yet brilliant style of comedy, reading his explanations to these ridiculous questions tickled some part of my brain that hadn’t been tickled in quite the same way before (only after writing this did I realize how terrible of a metaphor it was, but bear with me).

One of Randall Munroe’s webcomics (via xkcd.com) — **One of Randall Munroe’s webcomics** (via xkcd.com)

When I was this ten-year-old version of myself, I decided that there was one question I wanted him to answer more than anything: How fast would you have to throw a baseball at a bat to hit a home run? Five years later, I have a much better understanding of the process needed to answer a question like this, so I decided to have a crack at it. Far too many open tabs spread across multiple windows and one unorganized Google Sheets spreadsheet later, I have the semblance of an answer. What follows is a mix between Randall Munroe’s style and my own.

How Fast Would You Have To Throw A Baseball At A Bat To Hit A Home Run? Would The Bat And Ball Have To Be Indestructible?

Many statistics in baseball have risen and fallen over the years as indicators of overall hitting performance, one of the most recent and most useful being exit velocity. Exit velocity is very simple to understand, it’s just the speed of the ball as it leaves the bat. A higher exit velocity usually results in a higher likelihood to get a hit, and, more importantly for the question at hand, hit the ball over the fence. In the last five years, the exit velocity of the average home run has been around 103 miles per hour, according to Sports Illustrated, while the average ball in play is hit at around 89 miles per hour. This exit velocity is the result of two main factors: swing speed and pitch speed.

Data about pitch speed is widely available and relatively uniform (although that isn’t the case if you compare across eras), with the average MLB pitch speed coming in at around 93 miles per hour. Swing speed is trickier. While some information puts the average swing speed at around 90 miles per hour with top players swinging the bat over 100 miles per hour, that is measuring from the end of the bat, instead of the “sweet spot” where players actually hit the ball. Instead, the average swing is probably about 75 miles per hour, with players topping out around ten miles per hour faster.

Using data from the best MLB players, the worst MLB players, and average MLB players, we are able to create a formula that estimates the exit velocity from pitch speed and swing speed. With a normal pitch, a still bat would produce an exit velocity of 41.3 miles per hour. As 40 is often used to estimate bunts, this is perfect. With average conditions and an ideal launch angle, the ball would land between the pitcher’s mound and second base, although it would in practice be an easy fly ball. With a light breeze or a little luck, it might eclipse the 100-foot mark.

Needless to say, this is nowhere near our goal. The fastest pitch ever was 105.1 mph by Aroldis Chapman, but some estimate that because of where the ball was recorded in his era, Nolan Ryan broke 108 miles per hour. Even using the latter number, we add less than two miles per hour to the exit velocity, resulting in about eight more feet of distance.

What if we had Mother Nature (or a fancy gizmo) on our side? Baseballs travel farther in cold temperatures with high barometric pressures and powerful backwinds. Taking the most extreme instance of these ever recorded and plugging it into Alan M. Nathan’s Baseball Trajectory Calculator, we find that, despite the sustained 250 mile per hour backwind and perfect conditions, we would still need an exit velocity of 50 miles per hour for this to go over any major league fence, and even that’s a bit of a stretch; the ball would probably hit the wall instead of going over it. Even if this all worked in practice, it would require a pitch of 175 miles per hour.

Let’s say we don’t have an industrial-strength wind/cold/pressure machine (the scenario above is definitely cheating). Those 175 miles per hour suddenly don’t stretch anywhere near as far, with the ball probably landing on the infield dirt behind second base, although this too would be an easy catch. Without the swing, you simply need a lot more power on the pitch. We’re far outside the realm of possibility now. It’s hard to imagine that something resembling a human could throw anything that fast; even the world’s fastest pitching machine can muster a measly 143 miles per hour. Luckily, the Internet has an answer to everything: this time, it’s a supersonic baseball air cannon built by the Youtube channel SmarterEveryDay.

To hit our first true milestone, we need to get the ball up to an exit velocity of around 87 miles per hour, the number for the softest out-of-the-park home runs. These are all incredibly fluky home runs that usually result in fly balls, but it’s inside the realm of possibility. To get to an exit velocity that fast with a stationary bat, you would have to pitch the ball at about 525 miles per hour. That’s in the speed range of commercial airplanes (the diagram below is roughly to scale).

At this speed, the theoretical world is far removed from the practical one. When the ball hits the bat normally it compresses to somewhere between one-third and half of its normal size. When a baseball fired out of the air cannon mentioned above hit a steel wall while moving at around 1000 mph, it disintegrated. No, seriously. Even if the ball survived, it probably wouldn’t look like a baseball. The bat probably wouldn’t fare much better. Anything that gave it enough cushion to survive the impact would slow down the exit velocity. Maximize exit velocity, and the bat probably snaps. It’s a no-win scenario.

Getting up to a more reasonable exit velocity of 103, the average home run’s exit velocity takes some work. For the ball to leave the bat that quickly, we need to go transonic. The ball needs to be pitched at 675 miles per hour, or Mach 0.81. Anything faster than Mach 0.8 is considered transonic because some of the air around it is likely moving at or past the speed of sound. There’s almost no question that this would fail practically, but in theory, it would result in a home run.

How do we know this is right? Luckily, there are some data points we can check to see if our data. First of all, we have the measure of 40 for bunts that we used earlier and supports our formula. To further validate the data, though, we need to look at something called BESR. BESR is used to measure how fast the ball will bounce off of the bat. It’s essentially a ratio of how fast the ball comes in and how fast it goes out, factoring in both pitch and swing speed. Here is the formula:

Our real-world data suggests that the average major league bat’s BESR is somewhere between 0.53 and 0.55. The swing-less data is a little more generous, having the bat between 0.56 and 0.57. Given that bats are different, and we could find one of the better ones (because we don’t have to focus on length/weight), this is definitely reasonable and confirms that our formulas are reasonable to come up with theoretical data. If we opened this up to college stadiums and bats, maybe we could do it even slower, slow enough that the bat and ball wouldn’t break when they collided.

The supersonic air cannon from SmarterEveryDay probably marks the fastest we could move a baseball with current technology. It topped out at 1050 miles per hour. Plugging that into our formula, we find that the ball would leave the bat at 142 miles per hour, smashing the major league baseball exit velocity record. With the help of the backwind from earlier, the ball could easily beat the home run length record, perhaps launching the ball fully out of the park. If you think that’s a safety hazard, just wait until our pitcher gets a strike (looking, of course).

When shot at over 1000 miles per hour, nothing is stopping the ball. Nine leather gloves in a row weren’t enough. The ball wouldn’t even consider a truce with one catcher’s mitt, a set of pads, ribs, and the vital organs inside. I’m no health expert, but this would probably have a mortality rate of around 100%. If every catcher was American and the ball was pitched every year as many times as it was in 2018, then only 25% of pitches would have to miss the bat for “being ripped in half by a supersonic baseball” to be the third-leading cause of death in the United States.