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.
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
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
Ignore air resistance
No error in initial shooting velocity
Goes through center of hoop
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
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
- 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.
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