Jeremy Lin's Shot to Beat the Raptors

Two nights ago, Jeremy Lin of the New York Knicks hit a three-point shot to beat the Toronto Raptors.  Tied at 87, Lin was just behind the three-point arc and slightly right of center when he let go of the game-winner.  The ball fell through the net with just a half second remaining on the clock.  Click here for the story and video.  Lin has been in the news of late because of his great play, but also because of his great story.  It's not all the time that we see an undrafted player from Harvard making the NBA highlight reel!


I had fun analyzing Lin's shot.  After several timings, I estimate the ball's time of flight to be 1.32 s.  He looked to be just under 24 feet (7.3 m) from the basket, and let go of the ball at the top of his jump from a height above the floor of about 9.8 feet (3.0 m).  The basket sits 10 feet (3.0 m) off the floor.


To model the flight of the basketball, I included four forces.  The ball's weight of 22 oz (0.62-kg mass) is a downward force.  Because the ball displaces a volume of air equal to its own volume, the ball feels an upward buoyant force of about 0.31 oz (0.085 N), which is only 1.40% of the ball's weight.  In a direction opposite the ball's velocity is the drag force due to air resistance; the size of the drag force depends on the ball's speed.  The fourth force on the ball is the Magnus force, which is the same force that is responsible for curve balls in baseball and banana kicks in soccer.  Lin let go of the ball with backspin, so the Magnus force, which depends on the ball's speed and spin rate, has a component that is upward.  I estimate nearly three turns of the ball during its flight.


Using a computer to solve the ball's equation of motion that comes from Newton's second law, I get the trajectory in the image you see below (click on the graph for a larger image).

The ball left Lin's hand with a speed of nearly 19.9 mph (8.89 m/s) at 46.1° above the horizontal.  The ball's speed dropped to 17.5 mph (7.83 m/s) by the time it went through the basket.  Note that the ball reaches a maximum height of about 16.6 feet (5.05 m) above the court.


Below is a graph of the drag and Magnus forces on the ball as functions of time (click on the graph for a larger image).

Note that the forces in the above plot are at their minimum values when the ball is at maximum height, which is where the ball's speed is at its minimum value.  When Lin released the ball, the forces in the graph are at their maximum values because the ball's speed is greatest then.  The maximum drag force is 3.88 oz (1.08 N), which is 17.6% of the ball's weight.  The maximum Magus force is 1.30 oz (0.361 N), which is 5.90% of the ball's weight.  Note that the buoyant force, which I noted is 1.40% of the ball's weight, is nearly a quarter of the maximum Magnus force.


I hope we'll see more great shots from Jeremy Lin!