Basketball Jump Shot Physics

By | February 26, 2017

The math behind Michael Jordans legendary hang time Andy Peterson and Zack Patterson

Michael Jordan once said, quot;I don't know whether I'll fly or not. I know that when I'm in the air sometimes I feel like I don't everhave to come down.quot; But thanks to Isaac Newton, we know that what goes upmust eventually come down. In fact, the human limit on a flat surface for hang time, or the time from when your feet leavethe ground to when they touch down again,

is only about one second, and, yes, that even includes his airness, whose infamous dunk from the free throw line has been calculated at .92 seconds. And, of course, gravity is what's making itso hard to stay in the air longer. Earth's gravity pulls all nearby objectstowards the planet's surface, accelerating them at 9.8 meters per second squared. As soon as you jump,gravity is already pulling you back down.

Using what we know about gravity, we can derive a fairly simple equationthat models hang time. This equation states that the heightof a falling object above a surface is equal to the object's initial heightfrom the surface plus its initial velocity multiplied by how many secondsit's been in the air, plus half of the gravitational acceleration multiplied by the square of the numberof seconds spent in the air. Now we can use this equation to modelMJ's free throw dunk.

Say MJ starts, as one does,at zero meters off the ground, and jumps with an initial verticalvelocity of 4.51 meters per second. Let's see what happens if we modelthis equation on a coordinate grid. Since the formula is quadratic, the relationship between heightand time spent in the air has the shape of a parabola. So what does it tell us about MJ's dunké Well, the parabola's vertex shows ushis maximum height off the ground

at 1.038 meters, and the Xintercepts tell us when he took off and when he landed,with the difference being the hang time. It looks like Earth's gravitymakes it pretty hard for even MJ to get some solid hang time. But what if he were playing an away gamesomewhere else, somewhere faré Well, the gravitational accelerationon our nearest planetary neighbor, Venus, is 8.87 meters per second squared, pretty similar to Earth's.

If Michael jumped here with the sameforce as he did back on Earth, he would be able to get more than a meter off the ground, giving him a hang time of a little over one second. The competition on Jupiterwith its gravitational pull of 24.92 meters per second squaredwould be much less entertaining. Here, Michael wouldn't evenget a half meter off the ground, and would remain airbornea mere .41 seconds. But a game on the moonwould be quite spectacular.

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