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.

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:

b

adxx |sin|

(sin )b

ax dx

FREE THROW SHOTS

“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) = -32th’(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.

itth 216)(

i = 3.75 ft

75.3160 2 t

Arc Length of a Shot

 

Where dx/dt represents horizontal velocity

And dy/dt represents vertical velocity

t

dtdt

dy

dt

dxlength

0

22 )()(

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

716)( 2 vttth

7)738.0()738.0(1610 2 v

ftdttt

0

22 063.16)873.1532()325.20(

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.

)(16)( 2 tvtth

)9.0()9.0(160 2 v

ENDING VIDEO (CLICK VIDEO)

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>.