Modeling Swishing Free Throws

1
Modeling Swishing Free Throws

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

2
Disparity of Skill

• Isnt 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.

3
General Situation
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Overview of Model

• Determines desired shooting angle to shoot a
  swish from the free throw line
• Uses Newtons 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 (66) struggle with maintaining
  release angle, not initial velocity of the ball

5
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

6
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 )

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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

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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

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Determine Initial Velocity

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

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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

11
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

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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

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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

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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

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Front of Rim Situation

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

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a Function of Time

• Use Pythagoreans Theorem

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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

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Back of Rim Situation

• Condition to miss the back of the rim
• Only concerned with the time when the ball is at
  the rims height

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Excel

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

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Function to Select Desired Angle

• Example ball shot at 45 degrees

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Table of Rough Increments
Around 51 degrees appears to be the most
variation
Refer to handout for table
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Further Analysis

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

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My Best Shooting Angle

• 50.5º resulted in the best shooting angle

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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

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Questions?
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Bibliography

• Bamberger, Michael. Everything You Always Wanted
  to Know About Free Throws. Sports Illustrated
  88 (1998) 15-21.
• Bilik, Ed. 2006 Mens 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 .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.

27
Bibliography continued

• Metric Conversions. 2006. Accessed 12 September
  2006, from gth/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.