How Scientists See The Perfect Free Throw

A basketball computer program simulates millions of trajectories in search of the ideal shot.

Golden State Warriors Steph Curry. Photo by Ezra Shaw/Getty Images.

Some 20 years ago, my colleague Chau Tran and I developed a way to simulate the trajectories of millions of basketballs on the computer.


We went to the coaches and assistant coaches at North Carolina State University, where we are based, and told them we had this uncommon ability to study basketball shots very carefully.

Their first question was simple: “What’s the best free throw?” Should the shooter aim towards the front of the hoop or the back? Does it depend on whether the shooter is short or tall?

Math offers a unique perspective. It speeds up the amount of time it takes to see the patterns behind the best shots. For the most part, we discovered things that the players and coaches already knew — but every so often, we came across a new insight.

Simulating millions of shots

From a mathematical viewpoint, basketball is a game of trajectories. These trajectories are unique in that the ball’s motion doesn’t change much when it’s flying through the air, but then rapidly changes over milliseconds when the ball collides with the the hoop or the backboard.

To simulate millions of trajectories without the code taking too long to run, we tried any trick we could think of. We figured out how to go from modestly changing motion to rapidly changing motion, such as when the ball bounces on the rim or off the backboard. We learned how to turn large numbers of trajectories into statistical probabilities. We even created fictitious trajectories in which the ball magically passes through all of the physical obstacles (hoop, backboard, back plate) except for one, to see where it collides first.

How a mathematician sees a free throw. Larry Silverberg, CC BY-SA.

The free throw was the first shot that my colleague and I studied in detail. In close games, teams can win and lose at the free throw line. What’s more, the free throw is uncontested, so perfection in the free throw can pay off big. Top teams tend to shoot the free shot well.

Our program could tell us what chances the shooter had in sinking a free throw — and help us figure out what he was doing right or wrong.

Breaking down the free throw

We studied the free throw for about five years.

One of the first things we learned from our simulations and by watching TV footage, was that players with the same consistency can shoot free throws with anywhere from 75% to 90% accuracy. The difference was that the 90% players were being consistent at the right shot — the best trajectory.

The fate of a free throw is set the instant the ball leaves the player’s fingertips, so we looked closely at the “launch conditions” of the shot. The ball is located at some height above the floor. It has a rate at which it is spinning backwards (called backspin), and it has a launch speed and a launch angle. Since the shooter never launches the ball the same way, small differences account for a shooter’s consistency.

We found that about 3 hertz of backspin is the best amount; more than that does not help. It takes about 1 second for a ball to reach the basket, so 3 hertz equates to three revolutions in the air, from the instant the ball leaves the player’s hands to when it reaches the basket.

Next, assuming the player releases the ball at 7 feet above the ground, a launch angle of about 52 degrees is best. In that angle, the launch speed is the lowest, and the probability of the shot being successful is the greatest. At 52 degrees, the shooter can be off a degree or more either way without a large effect on the shot’s success.

However, launch speed is quite the opposite. It’s the hardest variable for a player to control. Release the ball too slowly and the shot is short; release it too fast and the shot is long. A player needs to memorize the motion of her entire body during release to impart the same speed consistently.

All else being the same, players who release from higher above the floor have a higher shooting percentage. That’s interesting, because our coaches at North Carolina State University and others I have talked say that taller players tend to shoot the free throw worse than shorter players do. It seems that the shorter players must try harder.

The last release condition was the most surprising: the aim point of the free throw. We found that the player should aim the ball to the back of the rim. Basically, the back of the rim is more forgiving than the front of the rim. At a release height of 7 feet, the gap between the ball and the back of the ring should be less than 2 inches. A small gap is best whether launching at low or high release heights.

Lessons learned

So what does this all mean for players out there aspiring to improve their free throw?

Our research suggests that players should aim the ball beyond the center of the rim. Launch the ball at a high angle and as high above the ground as possible. (The ball, at the highest point of its arc, should reach the top of the backboard.) Line up the ball to eliminate the side angle. And try to launch the ball with smooth body motion, to produce a consistent launch speed.

In the past few years, we’ve expanded our work to study where the best bank shots strike the backboard and developed a tool for anyone who wants to perfect it.

With tournament play approaching, I’m reminded of how competitive the game has become, and how it has truly become a game of inches. As an old basketball player, like many of you, I enjoy watching the game— and, every so often, catching a glimpse of that perfect free throw.

Sports
via Smithfly.com

"Seventy percent of the Earth is covered with water, now you camp on it!" proudly declares Smithfly on the website for its new camping boat — the Shoal Tent.

Why have we waited so long for camping equipment that actually lets us sleep on the water? Because it's an awful idea, that's why.

"The world is your waterbed," Smithfly says on its site. But the big difference is that no one has ever had to worry about falling asleep and then drowning on their waterbed.

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You could have to go to the bathroom in the middle of the night.

This guy.

It could spring a leak and you could drown while wrapped up in eight feet of heavy nylon.

A strong current could tip the tent-boat over.

There isn't any way to steer the darn thing.

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Mashable shared a charming video of the tent on Twitter and it was greeted with a chorus of people sharing the many ways one could die while staying the night in the Shoal Tent.

Oh yeah, it's expensive, too.

Even though the general public seems to think the Shoal Tent is a terrible idea, according to the Smithfly's website, it's currently sold out due to "popular demand" and it will be "available in 6-8 weeks." Oh, and did we mention it costs $1,999?

Lifestyle
via zoezimmm / Imgur

There are few more perniciously dangerous conspiracy theories being shared online than the idea that vaccines cause autism.

This has led to a decline in Americans vaccinating their children, resulting in a massive increase in measles. This year has already seen over 1,200 cases of measles, a disease that was eradicated in the U.S. nearly 20 years ago.

A 2015 Pew Research study found that 83% of Americans think the measles vaccine is safe, while 9% think it's not. Another 7% are not sure. But when you look at the polls that include parents of minors, the numbers get worse, 13% believe that the measles vaccine is unsafe.

There is zero truth to the idea that vaccines cause autism. In fact, a recent study of over 650,000 children found there was no link whatsoever.

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A great example of the lack of critical thinking shown by anti-vaxxers was a recent exchange on Facebook shared to Imgur by zoezimmm.

A parent named Kenleigh at a school in New Mexico shared a photo of a sign at reads: "Children will not be enrolled unless an immunization record is presented and immunizations are up-to-date."

This angered a Facebook user who went on a senseless tirade against vaccinations.

"That's fine, I'll just homeschool my kids," she wrote. At least they won't have to worry about getting shot up in school or being bullied, or being beat up / raped by the teachers!"

To defend her anti-vaccination argument, she used a factually incorrect claim that Amish people don't vaccinate their children. She also incorrectly claimed that the MMR vaccine is ineffective and used anecdotal evidence from her and her father to claim that vaccinations are unnecessary.

She also argued that "every human in the world is entitled to their own opinion." Which is true, but doesn't mean that wildly incorrect assumptions about health should be tolerated.

She concluded her argument with a point that proves she doesn't care about facts: "It doesn't matter what you say is not going to change my mind."

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While the anti-vaxxer was incorrect in her points, it must also be pointed out that some of the people who argued with her on Facebook were rude. That should never be tolerated in this type of discourse, but unfortunately, that's the world of social media.

Here's the entire exchange:

via zoezimmm / imgur


via zoezimmm / imgur


via zoezimmm / imgur


via zoezimmm / imgur


via zoezimmm / imgur


via zoezimmm / imgur


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via zoezimmm / imgur

The post received a ton of responses on imgur. Here are just a few:

"'In my opinion...' 'I believe...' That's not how facts work."

"You're entitled to your opinion. And everyone else is entitled to call you a dumbass."

"'What I do with my children is no concern to you at all.' Most of the time, true. When your kid might give mine polio, not true."

"If my child can't bring peanut butter, your child shouldn't bring preventable diseases."

It's important to call out people who spread dangerous views, especially how they pertain to health, on social media. But people should do so with respect and civility.

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