Modeling Swishing Free Throws

Michael LoneyAdvised by Dr. Schmidt

Senior SeminarDepartment of Mathematics and StatisticsSouth 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

00v

0v

0elow high

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 )

tvtx 00 cos)(

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

Tvl 00 cos

00 cos v

lT

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

tvgtty 002 sin

2

1

0

)(ty

0v

Determine Initial Velocity

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

,hTy

0000

2

00 cossin

cos2

1

v

lv

v

lgh

hl

glv

00

0 tan2cos

0v

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

0v

0

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

0oops0

oopsT0v

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

oopsT

g

ghvvT

oopsoopsoops

2sinsin 0

22000

oopsoopsoops TvTgh 00

2sin21

oops0

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

oops0

g

ghvvvTx

oopsoopsoopsoops 2sinsin

cos 022

00000

oops0

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

oops0

oops0

oopsTx

Front of Rim Situation

• (x,y) coordinates of center of ball and front of rim

s

a

b

tvgttv oopsoops00

200 sin21,cos

hDl r ,2

Rim ofDiameter rD

a Function of Time

• Use Pythagorean’s Theorem

hDl r ,2

2s

2

0022

002 sin21)2/(cos htvgtDltvts oops

roops

s

a

b

tvgttv oopsoops00

200 sin21,cos

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

22 2/bDts Ball ofDiameter bD

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

2/2/ rboops DlDTx

Excel

• Calculated initial velocity for any shooting angle

• Small intervals of time used and calculated both Front and Back of Rim Situations

• Determined and

0v

0

low high

Function to Select Desired Angle

• Example: ball shot at 45 degrees

000 ,min highlowe

0}4556,4545{45 e

Table of Rough Increments

45 46 47 48 49 50 51 52 53 54 55

0 0 0 1 1 2 3 2 2 1 10

)( 0e

• 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…

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

0v0

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.