Modeling Swishing Free Throws Michael Loney Advised by Dr. Schmidt Senior Seminar Department of Mathematics and Statistics South Dakota State University Fall 2006

Disparity of Skill Isn’t it annoying when you see NBA players making millions of dollars, yet they struggle from the free throw line? Only one-third of NBA players shoot greater than seventy percent from the free throw line.

General Situation

Overview of Model Determines desired shooting angle to shoot a “swish” from the free throw line Uses Newton’s Equations of motion which simulate the path of a projectile (basketball) Ignores sideways error, spin of the ball, and air resistance Assumes best chance of swishing free throw is aiming for the center of the hoop Assumes I (6’6”) struggle with maintaining release angle, not initial velocity of the ball

Derivation Process Shoot ball with fixed which will determine the initial velocity of ball to pass through center of rim (2 equations) Fix and vary the angle using two equations, and see whether the ball swishes by deriving two inequalities (Excel) After using inequalities, shooting angles and are inputs for function that determines the desired shooting angle

Horizontal Equation of Motion From physics Horizontal position of the center of the ball Will help determine the time when the ball is at center of rim ( l )

Time to Reach Center of Rim l is the distance from release to the center of rim T is the time at which the ball is at the center of the rim

Vertical Equation of Motion is the vertical position of ball for any t g is acceleration due to gravity (-9.8 m/s²) y(t) along with time T will help determine the initial velocity for any release angle to pass through the center of the rim

Determine Initial Velocity Set (time when ball is at the height of the rim) substitute T, and solve for

What Has Occurred Found time T at which ball is at center of rim Found initial velocity for the ball to pass through center of rim for any release angle For example: Shoot ball with 49º release angle resulting in an initial velocity ≈ 6.91 m/s

Shooting Error See what happens when player shoots with a larger or smaller release angle from Denote this new angle and note that this affects the time when the ball is at the rim height since still shooting with same New time called

Varying Times and Angles From Vertical Equation of Motion Solve for Function of and is the time at which the ball is at the height of rim

Horizontal Position of Ball From horizontal equation of motion Horizontal position of ball when shot at different angle (function of ) when at the rim height

Recap of oops Found time when ball passes through rim height when it is shot at Found horizontal position of ball when ball is shot at Must develop a relationship to determine whether these shots result in a swish

Front of Rim Situation (x,y) coordinates of center of ball and front of rim s a b

a Function of Time Use Pythagorean’s Theorem s a b

Guarantee a Swish Condition must be satisfied: Distance from center of ball to front of the rim (s) must be greater than the radius of the ball

Back of Rim Situation Condition to miss the back of the rim Only concerned with the time when the ball is at the rim’s height

Excel Calculated initial velocity for any shooting angle Small intervals of time used and calculated both Front and Back of Rim Situations Determined and

Function to Select Desired Angle Example: ball shot at 45 degrees

Table of Rough Increments 45 46 47 48 49 50 51 52 53 54 55 1 2 3 • Around 51 degrees appears to be the most variation • Refer to handout for table

Further Analysis Used Excel to further analyze shooting angles between 50 and 52 increasing by tenths of a degree Time intervals sharpened…

50.5º resulted in the best shooting angle My Best Shooting Angle 50.5º resulted in the best shooting angle

Further Studies Air Resistance: Affects 5-10% of path [Brancazio, pg 359] Aim towards back of rim ≈ 3 inches of room Vary both and by a certain percentage Shoot with 45º velocity ≈ 6.96 m/s and practice shooting at 50.5º release angle

Questions?

Bibliography Bamberger, Michael. “Everything You Always Wanted to Know About Free Throws.” Sports Illustrated 88 (1998): 15-21. Bilik, Ed. 2006 Men’s NCCA Rules and Interpretations. United States of America. 2005. Brancazio, Peter J. “Physics of Basketball.” American Journal of Physics 49 1981): 356-365. FIBA Central Board. Official Basketball Rules. FIBA: 2004. Accessed 12 September 2006, from <http://www.usabasketball.com/rules/official_equipment_2004.pdf>. Gablonsky, Joerg M. and Lang, Andrew S. I. D. “Modeling Basketball Free Throws.”SIAM Review 48 (2006): 777-799. Gayton, William F., Cielinski, Kerry.L., Francis-Kensington Wanda J., and Hearns Joseph.F. “Effects of PreshotRoutine on Free-Throw Shooting.” Perceptual and Motor Skills 68 (1989): 317-318.

Bibliography continued Metric Conversions. 2006. Accessed 12 September 2006, from <http://www.metric-conversions.org/length/inches-to-meters.htm>. Onestak, David Michael. “The effect of Visuo-Motor Behavioral Reheasal (VMBR) and Videotaped Modeling (VM) on the freethrow performance of intercollegiate athletes.” Journal of Sports Behavior 20 (1997) 185-199. Smith, Karl. Student Mathematics Handbook and Integral Table for Calculus. United Sates of America: Prentice Hall Inc., 2002. Zitzewitz, Paul W. Physics: Principles and Problems. USA: Glencoe/McGraw Hill, 1997.