And Michael Jordan Shoots...
This problem was found in an old workshop packet from the 1980's. I believe the worksheet was prepared by Robert Garvey, a mathematics teacher in the Louisville (KY) area. Though the problem is quite straight-forward mathematically, I am offering the problem because it might attract students interested in basketball, it suggests the use of simulation on graphics calculators, and it might create discussion relative to the accuracy of the simulation.
Michael Jordan shoots a jump shot from 20 feet straight out from the basket. The ball leaves his hand 8 feet above the floor with an initial velocity of 28 ft/sec and has a take-off angle of 60 degrees. Assume that the front rim extends one foot in front of the backboard, and that a shot arriving within a foot behind the backboard (as the 10 foot level) banks in for a score.
- Write a pair of parametric equations to simulate this problem and decide whether or not Michael Jordan scores.
- To make this problem more appealing construct a "basket" that gives a better visualization of this problem situation.
- Change velocity and angle settings slightly to see if this makes a difference. Does a velocity of 28.67 ft/sec score?
- Write equations to simulate a 30-foot shot.
Michael Jordan shoots a 20-foot shot. It leaves his hand at an angle of 60-degrees with an initial velocity of 29 ft/sec. The rim is exactly 3 feet below the top of the backboard and extends exactly 1.5 feet in front of the board.
- How high above the rim will the ball hit the backborad?
- At what angle will the ball hit the backborad?
- Will the ball score? (Ignore the diameter of the ball)
- Try velocities of 28.5 and 28.9 ft/sec to see if Jordan scores under these conditions, keeping an angle of 60-degrees.
Michael Jordan is guarding Larry Bird, who shoots a 20-foot shot. It leaves his hand at a 60-degree angle with an initial velocity of 28 ft/sec. The ball is 8 feet above the floor when it is shot. Michael is standing 2 feet in front of Bird with his hand stretching 8.5 feet above the floor. He begins a vertical leap as Bird releases the ball. Jordan leaves the ground at 15 ft/sec.
- What is the highest point of Jordan's reach?
- Does Jordan block the shot? (If his hand is higher than the ball at the 18-foot position, consider the shot blocked). If the shot is not blocked, how close does Jordan come?
- How soon before Bird shoots should Jordan begin his vertical leap to assure that he blocks the ball?
Hint: Before trying to write the simulation, some key elements to review are: parameter range, degree mode, and use of parametric equations.
Also, you might want to sketch out on paper an initial screen-layout of the ball, the shooter's hand, and the basket.
Solution Commentary: These hints were provided by Robert Garvey, assumed author of the problem...
- Set range as follows:
t-min = 0; t-max = 3; t-step=0.1
x-min = 0; x-max = 25; x-scl = 0
y-min = 0; y-max = 25; y-scl = 0
(Notice that the left side of the screen represents the point of departure of the ball, and the right side represents a place 5 feet behind the basket. The bottom of the screen is the floor and the top of the screen is the 25 foot ceiling.)
- To create a "basket" write these equations:
x(1t) = 18 + t/3; y(1t) = 10
x(2t) = 22 - t/3; y(2t) = 10
Before graphing this, what do you expect to show up on the screen? Explain the logic of the equations.
- To simulate the path of the ball, write these equations:
x(3t) = 28t(cos 60); y(3t) = 28t(sin 60) - 16t2 + 8
Can you explian the logic of these equations which trace out the x and y distances?
x(1t) = 20; y(1t) = 10 + 1.5t (draws backboard)
x(2t) = 20 - .75t; y(2t) = 10 (draws rim)
x(3t) = 29t(cos 60); y(3t) = 29t(sin 60) - 16t2 + 8 (path of ball)
Ranges are t: (0,2) step 0.1 and x: (0,25) and y: (0,25)
Hope these hints by Garvey work!