Monday, June 14, 2010

The Yang Way: Engineering a Jump Shot

by Yang Yang, Caltech class of 2009

Periodically, CxB3 presents posts by alumni and friends of Caltech Basketball. Mr. Yang graduated from Caltech in 2009 with a degree in Biology. While playing basketball for the Beavers, he also served as a contributing writer to The Tech, the campus newspaper.

"…basketball has become - at its best - the paramount synthesis in sport of intelligence, precision, courage, audacity, anticipation, artifice, teamwork, elegance, and grace."
- Carl Sagan, The Demon-Haunted World

Missing a shot during a basketball game is fairly common. With the exception of power forwards and centers who do their damage from within ten feet, most players shoot less than 50 percent from the field. Even great players struggle to make more shots than they miss. Michael Jordan only did that in six of the 15 seasons he played, finishing with a career average of 49.7 percent. To paraphrase Jeff Van Gundy, the NBA is a mostly miss league. If a circus knife thrower had that kind of accuracy… well, let's just say it wouldn't be a kid-friendly show. So why can circus performers fling knives 30 feet across the room to hit an apple above someone's head with high accuracy, but NBA players struggle to guide a basketball 30 feet across the court into a hoop?

Well, that's not a completely fair comparison. No circus knife thrower has to deal with a defender trying to intercept the knife or obstructing his vision. When dealing with pure shooting with no defense, basketball players are actually pretty amazing. Former Caltech guard Fred Newman holds the world record for 3-pointers made in a row with 209. Don't believe me? Check out the two-hour long video (cut version; uncut version, part 1). If that doesn't impress you, he also can make 88 free throws in a row while blindfolded. Still, whether it's Fred taking his 210th shot of that morning or Ray Allen taking a shot anytime during Game 3 of the 2010 NBA finals, players will miss.


Ask that question to an engineer, the answer will be mainly about suboptimal trajectories; ask a neuroscientist, it will be all about stability and precision of neural networks; and when you ask a psychologist, the reply will focus on mindset.

There are biological limits to accuracy - even when muscles are stimulated with the same voltage shock, there will be some variability in the output of force. When gauging distance, we will always make estimation errors.

To an engineer, shooting a basketball is just about solving equations. The mathematical laws which guide the trajectory of a basketball are well known. Lobbing an object into the air to hit a distant target has interested the military for millennia. Whether it's catapults, trebuchets, cannons or howitzers, armies for the past thousand years have used machines to bombard enemies from afar. It's a problem siege engineers have solved countless times in the past. In a simplified model, there are only two variables to consider - horizontal and vertical speed of the ball. Because we are constantly subject to the downward pull of gravity, we need an initial upward velocity to ensure the ball stays airborne. Once we know how long something can stay airborne from the initial vertical velocity, it's easy enough to calculate how fast its horizontal speed must be in order to reach the target. Introductory physics classes have tortured students with these problems for decades.

Unfortunately, accurately describing a basketball's motion requires much more detail than just the simple setup above. The collision of the ball with the rim will transfer some kinetic energy, depending on velocity and angle. This will affect the direction, speed and spin of the ball. With each collision, the spin of the ball will also influence all ball-to-surface contact, since it modulates the amount of friction between the two objects. In order to really describe real motion, we need a set of equations which considers all these factors and their simultaneous effects on each other. (A good mathematical summary can be found here). In fact, it is much easier to write a computer simulation which follow these rules and just run millions of iterations.

(to be continued)