Where to aim? Tran and Silverberg say you should aim for the back of the rim, leaving close to 5 centimeters – about 2 inches – between the ball and the back of the rim. According to the simulations, aiming for the center of the basket decreases the probabilities of a successful shot by almost 3 percent.
The engineers say that the ball should be launched at 52 degrees to the horizontal. If you don't have a protractor in your jersey, that means that the shot should, at the highest point in its arc to the basket, be less than 2 inches below the top of the backboard.
Free-throw shooters should also release the ball as high above the ground as possible, without adversely affecting the consistency of the shot; release the ball so it follows the imaginary line joining the player and the basket; and release the ball with a smooth body motion to get a consistent release speed.
"Our recommendations might make even the worst free-throw shooters – you know who you are, Shaquille O'Neal and Ben Wallace – break 60 percent from the free-throw line," Silverberg says with tongue firmly in cheek. "A little bit of physics and a lot of practice can make everyone a better shooter from the free-throw line."
The engineers used a men's basketball for the study; it is heavier and a bit larger than basketballs used in women's games. They also assumed that the basketball player doing the shooting was 6 feet 6 inches tall, and that he released the ball 6 inches above his head, so the "release height" was set to 7 feet. The free-throw line is 15 feet from the backboard, a cylinder-shaped opening that is 10 feet off the ground. Though it looks smaller, the diameter of a regulation basketball hoop is 18 inches; the diameter of a men's basketball is a bit more than 9 inches.