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Physics + MathPhysics & Math

Could the Physics of Free Throws Have Won the Cavs the Finals?

Basketball is a game of trajectories, which means there's a formula for the perfect free throw.

ByKelsey TsipisNOVA NextNOVA Next
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LeBron James, who will join the Los Angeles Lakers next season, with former teammate George Hill.

Last week marked a major shift in NBA history. On Sunday, LeBron James announced that he will leave his hometown of Cleveland, Ohio, for a four-year, $154 million deal with the Los Angeles Lakers. The move will dramatically reshape the NBA conference landscape for James, who has faced off with the Golden State Warriors in the past three consecutive NBA Finals—the last of which could have been won with a free throw.

At the end of Game 1, a free throw by Stephen Curry, who shot 94% from the free-throw line in the playoffs, put the Warriors a single point ahead of the Cleveland Cavaliers. But the Cavaliers soon had their chance to pull ahead. With 4.7 seconds left in regulation, LeBron James, who had an astonishing 49 of the Cavaliers’ 106 points at this time, passed the ball to teammate George Hill. A foul on the Warriors’ end sent Hill to the free-throw line for two foul shots.

If Hill makes one of the foul shots, the game is tied—two, and the Cavs win.

The Free Throw

The free throw is a kind of collective breath holding experience in basketball. The shot has sealed championships and broken other cities hearts. Entire professional players’ careers have been defined by their free-throw performance. Shaquille O’Neal was so notoriously bad at free throws that players purposefully fouled him, coining the term Hack-a-Shaq. O’Neal’s Lakers teammates once gifted him the book Free Throw: 7 Steps to Success at the Free Throw Line.

For fans, the shot looks deceptively simple. Its most discernable feature is, after all, that it’s “free:” the other nine players, the shot clock, and “biased” referees are all removed. But basketball, at its core, is a game of trajectories. Therefore, whether the 9.5-inch-in-diameter leather ball soaring 13 feet through the air eventually bounces off the 18-inch-in-diameter iron rim or settles smoothly into the net is largely determined by physics.

For Larry Silverberg, a mechanical and aerospace engineer at NC State University, solving that physics problem has become a bit of a passion project. Nearly twenty years ago, Silverberg, a dynamist by training and Cavaliers fan by choice, set out to mathematically map the perfect free throw in hopes that science could explain how some of the greats do it.

“Science explains the mystery, the magic, that superstars have already mastered, even though they often can’t explain it themselves,” says Silverberg. “And that way other people can try to emulate it.”

He started by distilling the shot into its four most salient features: release height, backspin, arc, and aim. With enough replications, Silverberg knew he could find what combination of these four features had the highest probability of success. So in between teaching classes on mechatronics and applied dynamics, he and a colleagues developed a computer software that simulated the trajectory of a million free throws.

“I’m an old guy who doesn’t play basketball anymore,” says Silverberg. “So now me and my colleague do this on the computer. It’s much less painful that way.”

The result—the perfect free throw, according to physics—was to release the ball 7 feet above the ground, at a launch angle of 52 degrees, with 3 hertz of backspin (about three and a half total rotations), aimed at the back of the rim.

At first glance, the findings seemed to confirm commonly held assumptions, like higher arc is better, but it also poked holes in some long-held teachings in free throw shooting. For example, college coaches that Silverberg talked to thought shorter players should aim for the front of the rim, assuming that aiming towards the back of the rim was reserved only for tall players.

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“And that’s actually not true,” said Silverberg. “Both tall and short should aim towards the back of the rim. You want there to be less than a 2-inch gap between the back of the ball and the back of rim.”

Perhaps most notably, the results showed that shorter players don’t have an advantage over taller players, despite the fact that in both the professional and collegiate basketball shorter players have better free throw stats.

In fact, height and free throw percentages seem to be negatively correlated in the NBA. Some of the greatest NBA centers of all time are also the worst free-throw shooters. Wilt Chamberlain, a four-time NBA MVP, averaged nearly 50 points a game, but only made 50% of shots from the line—the second worst percentage in the history of the NBA. Once a coach famously told Chamberlain: “If you were a 90% shooter, we’d never loose a game.”

Yet Silverberg’s findings show height is actually an advantage in shooting free throws. The reason being as simple as the ball doesn’t have to go as far through the air.

“You would think it would not make that much of a difference but 5’ instead of 7’ it is a couple of percent difference,” says Silverberg. “The shot is 13 feet from the basketball. Two feet is more than a 10% difference. Every little bit makes a difference.”

Silverberg also solved the age-old debate about who is better at free throws: men or women. Statistics from the NCAA show that the average free throw percentage for both genders has hovered around 70 percent for the last 30 years. But it’s long been thought that women had an advantage since they use a smaller ball.

However, the two balls are pressurized to the same level, which makes the smaller women’s ball bouncier, therefore less forgiving against the iron rim. Female players, on average, also tend to be shorter than their male counterparts—another disadvantage. Taking these factors into account, three years after they started the study, Silverberg and colleagues published a paper showing that female players in the NCAA are three percentage points better at free throws.

“No woman I have told that to is surprised by that,” Silverberg says. “They all say, ‘It took you 3 years to figure that out?’”

Game 1

Back at game 1 of the NBA Finals, George Hill of the Cavs is on the line. Hill is 6-feet 2-inches tall, and usually a high-70s free throw shooter. But this playoff season, he’s shooting 81% from the line. Hill dribbles the ball three times on the line, spins it in his hands, and releases. It goes in. The score is now 107-107. Hill stays on the line, ready for his next shot.

Game one of the NBA finals is akin to the New Hampshire primary in presidential elections; it sets the tone, foreshadows how well the teams are matched, and gives the victor the momentum. That is to say, there was more than Hill on the line.

More at Play Than Physics

Silverberg originally set out to map the physics of the free throw but in doing so proved that there is more than physics at play on the free throw line. So why are the players who are at a mathematical disadvantage better free throw shooters?

The answer is that they’ve developed something else: consistency

“It’s probably psychological,” says Silverberg. “It’s probably because if you’re short and you’re out on the basketball court, you have to try harder. The tall guys, they feel like they deserve to be out on the court. You never see a short player on the basketball court who doesn’t hustle.”

If hustle and grit explain why shorter players and female players fare better at free throws, which is more important: the physics (the form of the shot) or consistency (how much players practice)?

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There are two lines of reasoning. On one hand, knowing the best mathematical shot is important to know what to practice. If two players practice the same amount, the player who is consistent at the best shot may become a 90% shooter; the player who is consistent at a worse shot may still be a 70% shooter.

On the other hand, for players that need a lot improvement—the Wilt Chamberlain and Shaquille O’Neal types who shoot just 50%—a better strategy may be to switch to a new form entirely, one that is easier to get consistent.

A perfect example is NBA Hall of Famer Rick Berry. Berry was the greatest free-throw shooter in NBA history when he retired. He shot 90% from the free throw line. In one season, he missed only 9 free throw attempts. And he did so shooting underhanded, or “granny style.”

But Berry’s success is not because he cracked the code to the physics of the perfect form. The ball is released too low in underhanded shots, according to Silverberg’s simulations.

“The underhand shot is actually a worse shot, but it’s easier to be consistent,” says Silverberg. “If you look at person doing underhand shot, the motion of the body is a very smooth, single motion. You rock the ball and then you just release it. It’s one motion and very very simple.”

As a result, players can practice a more consistent shot. Silverberg has even recommended to some college coaches that shooters switch to underhand if the problem is consistency.

“No one has taken me up on that,” says Silverberg. “No one wants to do underhand. It’s very difficult to get players to do underhand. I think it’s the sissy thing. The guys feel like they’re sissies.”

This, it seems, has been a common occurrence for Silverberg. He’s talked to head and assistant coaches about his research, received feedback on what questions they want answered, tweaked the algorithms, but very rarely have the results changed anyone’s behavior. He admits he knows that there are investments with “bigger payoffs” than free throws, like, physical conditioning.

“We obviously do it fun because no one would ever pay us money to do this,” says Silverberg. “We’re not bitter that the world doesn’t center around this.”

Knowing the Score

Back at Game 1, Hill readied to take his second foul shot. Again, he dribbled three times, spun the ball, and released. This time, however, the ball ricocheted off the front of the rim and back into play.

Fortuitously, Hill’s teammate, JR Smith, got the rebound. For a moment, it looked like the Cavs may get a second chance. However, in what will go down in Cavalier infamy, presumably thinking his team was ahead instead of tied, Smith dribbled to half court to stall. The move sent the game into over time, where the Cavs would go on to lose 114-124. Seven days later they would lose the series.

Physics may be able to explain the shots, but sometimes just knowing the score helps too.

Photo credit: Erik Drost / Flickr (CC BY-ND 2.0)

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