The Physics of. . . Foul Shots

Why everyone in the NBA ought to be using a granny shot from the foul line

By Curtis Rist|Sunday, October 01, 2000

As a boy in Elizabeth, New Jersey, in the 1950s, basketball legend Rick Barry got some painful coaching lessons from his father, a semipro. While the youngster's friends liked to shoot their foul shots, or "free throws," in the respectable overhand style, the old man wanted Barry to toss them just as he did—underhand. "That's the way little kids shoot, and it didn't help that everybody calls it the 'granny shot,'" Barry says. "I didn't want any part of it, but my father drove me nuts until I tried it. And amazingly, it worked." Barry's average from the free-throw line bounced from 70 to 80 percent and kept on climbing when he became a pro. "Nobody ever teased me, but then it's hard to tease somebody when the ball keeps going in."

Judging by mechanics alone, this should be the case with just about every foul shot. "There's nothing simpler in basketball, because you can take all the time you want to make it, and there's nobody waving their arms in front of you trying to block you," says Peter Brancazio, a physics professor emeritus from Brooklyn College and author of SportsScience: Physical Laws and Optimum Performance. "It's like bowling. You do exactly the same thing over and over and over again." Yet while Barry can easily sink nine out of 10 shots, others fall far short. The late Wilt Chamberlain, for instance, could shoot a basket from just about anywhere on the court—except when he toed up to the line 15 feet from the hoop. There, the legendary "Big Dipper" sank barely five out of 10 shots, although he got plenty of practice: 11,862 attempted free throws, a National Basketball Association record that still stands. 

Sports columnists gripe about the bad free-throw techniques of modern players like Shaquille O'Neal, but no one has suffered more public humiliation at the free-throw line than the Knicks' Chris Dudley. One year, he made only three out of every 10 shots, and last season, when he managed to sink two free throws in a row during post-season play-offs, he made headlines ("Chris No Dud at Foul Line!" screamed the New York Daily News). "I'm convinced that from a physics standpoint, if everyone learned to throw underhand you'd see these statistics rise dramatically," says Brancazio. 

The key to a successful foul shot lies in the arc of the ball—in general, the higher the better. While an official-sized basket is 18 inches in diameter, the basketball itself is only about 9.5 inches, which gives a margin of 8.5 inches. But when the ball is thrown nearly straight at the basket—in the style of Shaquille O'Neal—the margin disappears because the rim of the basket, from the perspective of the ball, resembles a tight ellipse. "That's why these guys miss so much," says Brancazio. "Because of the sharp angle of the typical overhand throw, there ends up being a much smaller window for the ball to go in." If the ball comes down at the basket from a steeper angle—the way it does if tossed up in the high arc characteristic of an underhand throw—the margin reappears. "That means there's a far greater chance of making the basket," he says.

Using lots of trigonometry, Brancazio calculated the optimal angle of the arc from the free-throw line. If tossed at 32 degrees or less, the ball will likely hit the back of the rim. "That doesn't mean it won't go in, but it will certainly bounce off the metal and reduce the chance of success," he says. At angles greater than that, the ball has a chance of making a nice swish. The optimum angle, he calculated, is 45 degrees—plus half the angle from the top of the player's hand to the rim. "The shorter you are, the steeper that angle has to get to give you the best chance of making the shot," he says. Of course, lobbing a ball very high so that it comes down nearly straight into the basket would be the most efficient technique, but a shot like that "is almost impossible to aim," says Brancazio. Instead, he says, his formula makes it possible for a player to shoot with the largest possible margin for error.

Another factor of the granny shot also helps a free throw win cheers rather than jeers: a backward spin added to the ball. If a ball with backspin happens to hit the metal rim of the basket, the friction of contact suddenly reduces its forward velocity. "It's like a drop shot in tennis—the ball bounces, but it doesn't have a forward motion on it," says Brancazio. This effect tends to freeze the ball at the rim and greatly increases the chance that it will tip into the basket rather than ricochet off. 

The underhand throw can also minimize the drift of the ball. "A little sideward movement at the start of the throw will translate into a big movement toward the end," says Tom Steiger, an assistant professor at the University of Washington in Seattle who teaches basketball physics in a sports science class. The trick to keeping the ball moving along a single plane toward the basket lies in "minimizing the x-axis motion," he says. "In other words, you have to keep your elbows tucked in." If they're sticking out, that can easily add an unwanted nudge to the ball, which results in a missed shot. The underhand throw provides better stability than the overhand "because you're holding the ball with both hands," Steiger says. This helps players balance the subtle motor muscles in the hands, and keeps them more relaxed. The movement of the underhand throw is a simple, easy-to-control upward pendulum motion. By contrast, the more conventional overhand free-throw shot involves separate movements of the wrist, elbow, and shoulder that can add errors. "If the ball ends up rolling off one side of your hand even a little bit, you'll miss," he says.

Despite the logic of a granny approach to foul shooting, no NBA player has used it since Barry retired in 1980. "That baffles me," Barry says. "With the underhand shot, I could make 80 percent of my throws with my eyes closed. And I do mean closed." Over the years he has tried to convert everyone from four of his sons who have played professionally to O'Neal to Chris Dudley—but nobody paid any attention. "A lot of guys who are lousy at the free throw would be prime candidates for this, but they just won't do it," says Barry, who for the last five years coached a minor-league team called the Florida Sea Dragons. "I mean, how can guys call themselves professionals when they can't even make 60 percent of their free throws? Where's their sense of pride?"

Perhaps that's the problem. "As good as it is," Steiger says, "it does look kind of stupid."

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