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## The Calculus of Basketball

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**The Calculus of Basketball**Hindu Rao Adi Gulati Matthew Shadish**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.**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.**Positions**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)**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**• 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**Assumptions: Ignore air resistance No error in initial shooting velocity Goes through center of hoop**Components Of Shots**• shooting angle • speed of ball**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 Cont.**• Best angle is between 45 and 60 • Start solving: • Substitute t in =**Shaq Example:**• 7 foot or 2.13 m + .45 m for arm length = 2.58 m v=7.069 m/s**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**• Where dx/dt represents horizontal velocity • And dy/dt represents vertical velocity**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.**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**- 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**"Basketball." Wikipedia. 27 May 2009 <http://en.wikipedia.org/wiki/Basketball>. 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 <http://isolatium.uhh.hawaii.edu/m206l/student/shaq/shaq.htm>. Laughead, George. "History of Basketball." Kansas History. 5 Jan. 2005. 26 May 2009 <http://www.kansasheritage.org/people/naismith.html>. "Player Surveys." National Basketball Association. 27 Nov. 2007. 26 May 2009 <http://www.nba.com/news/survey_height_2007.html>.