The Physics of a Souvenir

I first met Larry in graduate school at Berkeley. He was one of my fellow slackers sneaking out of our basement laboratories on Sundays. We’d take BART through the East Bay to the Oakland Coliseum. For a couple of bucks, we’d plop ourselves down in the right field bleachers and enjoy a doubleheader.

Larry was a Mets fanatic and taught us the ways of fandom—East Coast style. Now Larry was a little guy, maybe 5-foot-5 and 120 pounds, but somewhere in that slight frame he possessed a bellowing voice. From the moment the visiting team took the field Larry began his daylong riding of the right fielder, whether it was Ken Singleton, Bobby Bonds or Reggie Jackson. They got an earful all afternoon.

It didn’t matter the inning, the score, or even what they did at the plate. If Reggie just struck out: “Hey, nice whiff, you bum!” or Bonds hit a base clearing two bagger: “Hey, double or not, you’re still a bum!” Of course, if Singleton made an error Larry went off: “Why do you bother to wear a glove? You bum!” “Where did you learn that trick? You clumsy bum!”

In addition to calling the right fielder a bum two hundred times a game, Larry had a million one-liners for various events that occurred during the contest. Unfortunately, all but one of those quips is now lost to the amyloid plaques building in my brain.There is no definitive answer as to who coined the phrase, “a souvenir for some lucky fan,” but we do know that Larry had his own variation. Whenever a line-shot would scream over the dugout and into the stands, Larry would chirp, “There’s a concussion for some lucky fan!” But is it? Let’s look at it through the lens of physics and find out.

From Statcast, we know that the ball can leave the bat at speeds approaching 120 mph. Of course, it  begins to slow down immediately due to the drag force as it travels through the air. You know about the drag force. For example, you understand skydivers slow down due the upward drag force the air exerts on their parachutes. You may have even felt the drag force directly by sticking your hand out the window of a moving car.

Let’s use some reasonable numbers for that screamer over the dugout. Since the ball is foul, the collision with the bat was not a direct hit. So the ball probably left the bat at something less than 120mph…100 mph let’s say.

In Comerica Park, it is about 100 feet from home plate to the seats behind the center of the Tigers’ dugout. So, let’s do the physics. The ball will get to “some lucky fan” in about three quarters of a second and it will be moving at roughly 82 mph. Batters and catchers wearing helmets get concussions from 90 mph pitches, so an unprotected fan will certainly be in the same unfortunate situation or worse. Some souvenir.

That bullet over the dugout is so scary, so let’s think about a foul pop-up instead. The physics you learned in high school would tell you that a ball heading upward at say 90 mph will also land at 90 mph. This is clearly wrong, because typically, fans in the seats behind the dugout will pray they have enough time to move out of the way of a line drive, but they will eagerly try to catch a falling pop-up.

Your high school physics forgot to take into account the drag force. While the pop-up rises, the drag slows it down. In addition, while the pop-up drops, the drag force also slows it down. So, a pop-up headed for a fan behind the dugout that leaves the bat at 90 mph will land at a little bit above 50 mph. Enjoy this catch by a young Tigers fan.

Things get even easier (and safer) for fans in the second deck. As the pop-up rises, gravity and the drag force both slow down the ball. If “some lucky fan” happens to be able to catch the ball right at the peak of its flight, the ball will be moving at only about 10 mph. That’s why you see more good plays by fans higher up in the stands – the ball is moving much more slowly.

What about the best souvenir of all, a home run ball? On May 25, 2015 Miguel Cabrera hit a line shot to left field off Houston’s Luke Gregerson. According to Statcast, it came off the bat at 111 mph—his highest batted ball speed for a homer this season. The ball traveled 401 feet. Here’s the video.

This ball probably landed with a speed around 55 mph, well within the ability of a typical fan to grab. In fact, a week later a young Tigers fan caught a similar line drive grand slam. Unfortunately, it was hit against his team. He can be seen crying in the video, but gave a great interview once he regained his composure.

Miggy is not known just for line drive homers. He hits his share of the majestic fly ball variety. On May 16, 2015 he hit the 400th career round-tripper of his career off Tyler Lyons of the Cardinals. Statcast tells us the speed off the bat was 106 mph and the distance was 417 feet. Here’s the video.

While the ball was hit to dead center where no fans could attempt to catch it, the speed it had when it hit the grassy mound was a bit over 50 mph. So, regardless of the trajectory of a homer, it will land at between 50 and 55 mph—about the same speed pop-ups have in the field level seats.

Well Larry—wherever you are—thanks for all those sunny Sundays at the ballpark and the idea to think about the speed baseballs have as they reach the stands. Here’s hoping your souvenirs have always been pop-ups or homers and not line-shots over the dugout. You bum!