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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.