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The Calculus of Basketball. Hindu Rao Adi Gulati Matthew Shadish. Intro Video (Click Video). Introduction. Court Length: 94 ft. Court Width: 50 ft. Rim Height: 10 ft. 3-Pt line distance from center of basket: 22 to 23.75 ft. Free-throw line distance from backboard: 15 ft.

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the calculus of basketball

The Calculus of Basketball

Hindu Rao

Adi Gulati

Matthew Shadish

  • Court Length: 94 ft.
  • Court Width: 50 ft.
  • Rim Height: 10 ft.
  • 3-Pt line distance from center of basket: 22 to 23.75 ft.
  • Free-throw line distance from backboard: 15 ft.
history fun facts
History & Fun Facts

In 1892, James Naismith published his rules for the game of "Basket Ball" that he invented.

Dribbling was not part of the original game, but was introduced in 1901.

Coaches allowed to address players during time-outs from 1949.

The shot clock was first introduced by the NBA in 1954, to increase the speed of play.

Goaltending became a violation in 1944.

From 1924, players that received a foul were required to shoot their own free throws.

The NBA added a third official in 1988.


Point guard: organizes the team's offense by controlling the ball

Shooting guard: creates a high volume of shots

Small forward: mostly responsible for scoring by cuts to the basket and dribble penetration

Power forward: plays offensively with back to the basket

Center: uses size to score

player stats for 2007 2008 season
Player Stats (For 2007-2008 Season)

Average Height: 6 ft 6.98 inches

Tallest: Yao Ming at 7 ft 6 inches

Average Weight: 221 pounds

Heaviest: Shaquille O’Neal at 325 pounds

Average Salary: $5.356 million per year

Highest: Kevin Garnet tops the chart at $22,000,000

Average Experience: 4.71 years

Most Experienced: Dikembe Mutombo at 16 years

displacement distance traveled
Displacement/Distance Traveled
  • Average distance a player travels in a game: About 5 miles
  • A model for displacement:
  • A model for distance traveled:
nearly nothing but net shots
“Nearly Nothing but Net” Shots


Ignore air resistance

No error in initial shooting velocity

Goes through center of hoop

components of shots
Components Of Shots
  • shooting angle
  • speed of ball
components of shots cont
Components Of Shots Cont.
  • Length of a shot with initial speed Vo and shooting angle:
    • Horizontal component:
      • X=(Vo)(cos(a))(t)
    • Vertical component:
components of shots cont1
Components Of Shots Cont.
  • Best angle is between 45 and 60
  • Start solving:
  • Substitute t in


shaq example
Shaq Example:
  • 7 foot or 2.13 m + .45 m for arm length = 2.58 m

v=7.069 m/s

bouncing a ball
Bouncing a Ball

Where i is the initial height of the ball.

t = 0.484

 When dropped from a height of 3.75 ft, the ball takes approximately 0.484 sec to hit the ground.

 h’(t) = -32t

h’(t) = -32(0.484) = -15.492 ft/s

When dropped from a height of 3.75 ft above the ground, the basketball hits the ground at a velocity of about 15.492 ft/s.

i = 3.75 ft

arc length of a shot
Arc Length of a Shot
  • Where dx/dt represents

horizontal velocity

  • And dy/dt represents

vertical velocity

average arc length of a free throw shot
Average Arc Length of a Free Throw Shot
  • distance from the free throw line to the basket is 15 ft
  • distance from the ground to the basket ring is 10 ft
  • average NBA player height is 6.583
  • assume basketball leaves hands at 7 ft
  • assume the time the ball takes to reach basket ring is about 0.738 sec
average arc length of a free throw shot cont
Average Arc Length of a Free Throw Shot Cont.

dx/dt = 15/0.738 = 20.325

 - to find dy/dt, we use this equation

Where v is the upward velocity of the ball

 v = 15.873 ft/s

dy/dt = h’(t) = -32t+15.873

 Average arc length of a free throw is thus 16.063 ft

vertical jump
Vertical Jump

- average vertical of an NBA player is 28 in

- air-time of a 28 in jump is about 0.9 sec

We can use this equation to find the upward velocity, v, of a player

  v = 14.4 ft/s

Average upward velocity produced by a player when jumping is about 14.4 ft/s

 - to find the upward velocity of the player at any given time, we take the derivative of h(x)

 h’(t) = -32t + 14.4

 - to find point at which the player is at his

highest, we set h’(x) equal to 0

 0 = -32t + 14.4

t = .45

 Player reaches maximum height of vertical jump at 0.45 sec.

works cited
Works Cited

"Basketball." Wikipedia. 27 May 2009 <>.

Gablonsky, Joerg M. "Modeling Basketball Free Throws." Editorial. Our First Model (2005): 1-9.

Kasch, Niels. Please help Shaq somebody??? 21 Apr. 2004. 26 May 2009 <>.

Laughead, George. "History of Basketball." Kansas History. 5 Jan. 2005. 26 May 2009 <>.

"Player Surveys." National Basketball Association. 27 Nov. 2007. 26 May 2009 <>.