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Physics + MathPhysics & Math

You Shot a Rubber Band off Your Thumb. Why Didn’t Your Thumb Get Hit?

The answer, it turns out, involves some pretty serious physics.

ByKatherine J. WuNOVA NextNOVA Next
iStock-147333066.jpg

Want to shoot a rubber band off your thumb? Here's how to pick the best elastic for the job. Image Credit: iStock

Whether out of mischief or just plain boredom, most of us have fired a rubber band slingshot at some point. It’s a pretty thoughtless task: Hook a rubber band around your thumb, stretch it backwards with your other hand, and fire.

If you’re successful, it takes just a couple milliseconds for that elastic to rocket off the tip of your finger and commence its arc across the room (perhaps eventually pinging off a wall or the arm of a hapless victim). But occasionally, the endeavor ends with a rubber band smacking up anticlimactically against the back of your thumb—or paperclip, like in this video below:

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It probably sounds like pure chance. But it turns out there’s actually some serious math that goes into determining whether or not your thumb will get in the way. And Alexandros Oratis and James Bird, who study fluid dynamics at Boston University, have done the legwork (or fingerwork) to figure out what keeps your biggest digit out of harm’s way.

“This work is really fun,” says Severine Atis, a biophysicist at Harvard University who was not involved in the study. “Everyone has played with these elastics… if they wonder how this works, this paper gives a really neat answer!”

Let’s start at the beginning. In order to hold the front of the rubber band in place as the rest is drawn back into a slingshot, your thumb has to keep steady. Even though it remains still, that finger is pushing against the front of the rubber band to counteract the elastic’s backward tug. Essentially, the rubber band and your thumb are shoving up against each other like two very well-matched arm wrestlers. When the rubber band is shot forward, however, the tension from the elastic abruptly disappears, releasing the thumb from this static stalemate—as if one of the two arm wrestlers has suddenly gone slack. With nothing holding it up, the thumb deflects forward. And if the thumb is able to shimmy out of the way before the back of the elastic reaches it, the rubber band proceeds unimpeded. For this to happen, though, the front of the rubber band has to liberate the thumb before the back catches up.

That’s Oratis’ hand in the video. Apart from showing off his one-off stint in hand modeling, the video is actually pretty revealing. When the elastic fires, it moves in two subtle, but different, waves. The one that’s most obvious is the rear end of the rubber band—the part that you don’t want hitting your thumb. Fortunately, the back of the rubber band trails behind behind a separate wave that ripples more quickly through the rest of the elastic.

rubber band time lapse

A time-lapse image that shows a rubber band rocketing off study author Alexandros Oratis' thumb. The dotted line in the righthand panels shows the movement of the thumb as it deflects forward, out of the path of the back end of the rubber band to its left. Image Credit: James Bird and Alexandros Oratis, Boston University

This faster wave releases the elastic from tension: Everything it passes immediately goes slack, while the length of elastic ahead of it remains taut. Meanwhile, the rear wave lags behind, dragging the tail of the rubber band.

Essentially, the faster wave carries the information that the rubber band has been released—and when this wave reaches the thumb, this digit is immediately catapulted into deflection. Because the slow wave controlling the back of the rubber band remains in the shadow of the fast wave that releases the thumb, there’s a slim window of opportunity for the thumb to evade the elastic.

To maximize that window, there are a couple things to take into account. First, you can give the thumb’s deflection a little extra oomph by using a thick, wide rubber band that’s got a big circumference; this basically makes the thumb work harder to hold the elastic in place, priming the digit to spring forward when it’s set free. (Technically, you could also up your chances by reducing the size of your thumb… but let’s assume this won’t necessarily be under your control, unless you’ve somehow got a couple different sets of hands to choose from.)

Your thumb also ends up getting a bigger workout if you stretch the elastic more—but the excess tug on the rubber band can create a second effect as well. The more the elastic is pulled backward, the faster the rear of the rubber band will hurtle toward your thumb, narrowing that critical window between the two waves in the elastic. This brings us to our second strategy: While it’s important to get the rubber band taut enough to make it fly, you actually want to avoid over-straining the elastic.

If all this sounds complicated, that’s because it is. Since the stretch of the rubber band needs to fall into something of a sweet spot, it may actually be more practical to focus on using a thick, long, wide rubber band. In their study, which publishes tomorrow in the journal Physical Review Letters, Oratis and Bird actually went through the trouble of assessing just how important the thickness of a rubber band is to this deceptively simple process. (Spoiler: It matters.)

“There’s so much in this world we take for granted,” Bird says. “We’re surrounded by elastics, to the point that they’re more or less unnoticed… it’s wonderful to pause for a moment to recognize the beauty in the mundane.”

Oratis and Bird even came up with an equation that can tell you how to definitively avoid failure to launch. Here it is:

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Okay, that probably wasn’t all too helpful. There’s a lot going here, but really, this equation is just taking into account the factors that affect the thumb and the rubber band, which are the only things at play. Partitioning the equation out, we end up with:

equation 2.jpg

r, d, and ρ/ρt represent the radius, length, and density of the thumb, respectively. ɑ represents how much the thumb needs to move out of the way to avoid getting struck by the rubber band.

ℓ0, b, and h account for the length, width, and thickness of the rubber band, respectively.

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Summing up what we know, it makes sense that we’d want to keep the value of the left half of this equation small (less than 1). That means we want the top half of the equation—the stretch of the rubber band and the size of the thumb—to be smaller than the bottom half, which factors in the size of the rubber band.

So assuming you’re not out to bruise your thumb, it’s probably best to select a hefty rubber band—and avoid pulling it back too far, Oratis advises.

On the other hand, “someone with a big thumb and a short, thin elastic will probably get hit,” Atis says. But at the same time, she adds with a laugh, that brawny thumb is probably better built to handle the sting.