March Madness Math
March Madness, the annual NCAA basketball playoff spectacle in which millions of us, firmly docked in front of the TV screen, consume 1,000 calories an hour while watching young athletes burn 12 calories a minute, began this week.
If you’re planning to participate in this national sit-in, you can drastically enhance the viewing experience by pondering the parabola.
It’s the elegant arched trajectory naturally formed by any projectile, from an artillery round to a tomato, moving in a gravitational field. Parabolas have been extensively studied since people started throwing stuff at each other, and they shape the outcome of many ballistic sports, such as baseball, golf, football, shot put and more. But they reach their apex in basketball, where field goals and free throws demand precision control of parabolas.
But not just any parabola. Success favours a fairly high arch. The ball must pass through the hoop with a little room to spare, and that limits the possibilities. The hoop is roughly 48 centimetres in diameter, and the men’s ball is about 24 centimetres wide (women’s about 23.5). So if the men’s ball were thrown straight down from above that is, at an angle of 90 degrees to the horizontal hoop rim, as in the classic Michael Jordan airborne dunk, there would be just under 11 centimetres of free space all around, a comfy margin.
But as the angle decreases and approaches the horizontal, the free space for a “nothing but net” shot gets much smaller. At 55 degrees, it’s about 6.4 centimetres. At 45 degrees, it’s down to 3.8 centimetres. And at 30 degrees, it’s basically impossible to get the ball straight into the basket, even with a full scholarship and more tattoos than a Hell’s Angels convention.
Not surprisingly, increasing the height at which the player launches the ball not only reduces the distance to the basket, but raises the entry angle of the ball’s parabolic arch, allowing more free space.
In a 1980s study, Peter Brancazio, a Brooklyn College physics professor, determined that adding two feet to the height at which a shot leaves the player’s fingers increases success by a whopping 17 per cent. No wonder you see so many jump shots.
But is there a launch angle that gives the maximum probability of a perfect telegenic swish? Well, there are many different parabolas that will do the job, and the choice varies according to player height, personal preference and court position.
But one way to decide, Brancazio wrote 25 years ago in Sport Science: Physical Laws and Optimum Performance, is to “consider the amount of force needed to launch the shot.
“It is to the shooter’s advantage to use as little force as possible,” he reasoned, because the less the force, “the more quickly and effortlessly (the ball) can be released.”
Okay, fine, but how do we know what takes the least force?
Here, physics comes literally into play.
We know from theory and experiment that you get the most distance with the least effort by firing a projectile at 45 degrees, exactly midway between vertical and horizontal. And we can assume that least-effort shooting is really important for a player taking a jump shot, because he or she can’t push against the floor for power, especially in heavy defensive traffic. So the fastest and easiest angle would seem to be 45 degrees.
Except when it isn’t, which is a lot of the time. The reason is that 45 degrees is the ideal least-effort angle only if the ball is shot from the same height as the basket, which is 10 feet above the floor. So it’s perfect for a 7-foot player whose arms reach two feet over his or her head, and who jumps a foot off the floor to shoot.
The rest of us will be launching the ball “uphill” (that is, as if we were firing a cannon at a target on a higher elevation). So we’ll need larger angles.
How much larger? Again, science comes to the rescue. Brancazio explains that you need 45 degrees, plus half the angle formed by a straight line between the position of the ball at launch and the basket. Depending on your height and where you are on the court, that typically ranges from 7 to 14 degrees.
Thus, for a shot leaving your hands at eight feet above the floor from 18 feet out, you’ll want to launch the ball at a bit more than 48 degrees. For most players at a distance of 10 to 25 feet, the least-effort angle ranges between 47 and 52 degrees.
Using that system, you can calculate the ideal free-throw angle. It’s 13.75 feet from the free-throw line to the centre of the basket, and a 6-foot player launches the ball from about seven feet above the hardwood. That works out to a shooting angle of 51 degrees.
Of course, Brancazio did his calculations long before the advent of the modern computer. But a new state-of-the-art study gives basically the same result. Last November, engineers at North Carolina State University published an analysis of hundreds of thousands of 3-D computer simulations of free throws. Their optimal angle: 52 degrees. (Check it out during the playoffs. Seen from the side, a 52-degree free-throw parabola has its highest point just about even with the top of the backboard.)
Free-throw success is also improved by adding a little backspin, which pushes the ball downward if it hits the back of the rim.
The North Carolina State engineers calculated the ideal rate of free-throw backspin at three cycles per second. That is, a shot that takes one second to reach the basket will make three full revolutions counterclockwise as seen from the stands on the player’s right side.
While you’re at it, stop to remember Menaechmus, the geometer who first described the parabola in the 4th century B.C. He never made a layup, but he had game.