modeling swishing free throws michael loney advised by dr. schmidt senior seminar department of...

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.