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Original Issue


Kent Hannon's article An Idea That's Gotten Way Off the Ground (Nov. 28) on improved field-goal percentages in college basketball was read with great interest by the advanced placement physics class at Chats-worth (Calif.) High School. His analysis of the phenomenal shooting of today's player was appreciated by all. Unfortunately, if a player starts with Professor Enoch Durbin's range formula, the result will be an "air ball." The proper expression is

r = Vo[2] sin 2θ/g

However, even the correct equation will give the range only if the ball takes off and lands at the same height and if there is no air resistance. Hence, the ball would have to be released at the height of the rim in a vacuum—a condition that would surely have a disastrous effect on spectator health. If the gym were not evacuated, other factors, such as the rotation of the ball, would have to be considered.

We do agree with Hannon that shooting a basketball is not something that can be reduced to a mathematical expression. If it were, the NBA would be stocked with Caltech grads.
Chatsworth High School
Chatsworth, Calif.

Professor Durbin's equation should read

R(range) = V[2](velocity)/G(gravity) sin2θ

where θ is the angle from which the basketball is shot. This equation assumes that the ball is shot from the same height as the hoop, there is no air drag, the ball is not rotating, the ball is not banked off the glass or slam dunked.

This type of oversimplified analysis led some scientists to believe that a curve ball was an optical illusion.

Using your equation, the maximum range would be achieved by shooting the ball at a 90-degree angle (i.e., straight up). Better stick to sports!
Gettysburg College
Gettysburg, Pa.

Alas, 2 sin θ does not equal sin 2 θ.
School of Engineering
Columbia University
New York City

My students and I have enjoyed reviewing this formula to find out why SPORTS ILLUSTRATED had it wrong. I commend the photographer for the two shots on page 42, however. These pictures are excellent examples of motion along a parabola.
Associate Professor of Mathematics and Computer Science
University of Tennessee-Martin
Martin, Tenn.

We are sure that the equation presented in the article is merely an oversight.
Texas A&M University
College Station, Texas

•SI goofed. For the correct formula and Professor Durbin's analysis of the problem, see below.—ED.

The question is: Is there a preferred way to shoot a basketball based on the physics of the process? The answer is yes! One can analyze the problem using a purely ballistic calculation that assumes a clean shot and neglects air friction, opposing players and bouncing off the rim or backboard.

There are three relevant equations. The horizontal distance the ball travels from shooter to basket, or range (R), is given by

R = v[2] sin2θ/G

where v. is the launch velocity. G is the acceleration due to gravity and θ is the angle of launch as measured up from the horizontal (90 degrees is straight up).

The second equation

ΔR = 2v[2] Δθ cos2 θ/G

shows the change in range (ΔR) due to an error in launch angle (Δθ).

The third equation shows the change in range due to an error in launch velocity (Δv)

ΔR= 2v Δv sin2θ/G

From these equations we draw the following practical conclusions:

1) The minimum throwing velocity required for any range occurs at a launch angle of 45 degrees (sin 2θ=1, its maximum value). Higher or lower launch angles require higher launch velocities. The lower the launch velocity, the lower the velocity of the ball when it arrives at the hoop, which means the lower the bounce off the rim when you miss a clean shot. If the object were to throw the ball the farthest, the 45-degree launch angle also would result in the longest range for a given velocity.

2) The minimum change in range due to a small error in launch angle occurs when Œ∏ = 45° (cos 2Œ∏ = 0). Therefore, if you shoot at 45 degrees, your angular error causes the least range error. However, if you make an error in either direction around 45 degrees, the ball range is shortened. Further, as the launch angle is increased, the effective hoop cross section gets larger. The ball is coming in from a steeper angle. Thus, any errors in launching the ball should be on the high angle side. My suggestion would be to launch the ball at 48 degrees.

3) The third equation teaches us that the error in range is smallest when the launch velocity is lowest, since the range area is proportional to Δv times v.

If you study angles of launch at a basketball game, you will notice that the most accurate shooters are those who shoot at an angle slightly higher than 45 degrees. I am certain that they have found this angle without solving differential equations. But I would suggest that coaches paint a 48-degree line on the wall in order to show players who are having accuracy difficulties what this angle looks like.

Incidentally, if you are a quarterback trying to throw the bomb, or a kicker trying to kick a field goal from halfway down the field, these same calculations show that 45 degrees is your angle. If you are an outfielder trying to throw the ball to home plate, 45 degrees will permit you to throw it the farthest, but not necessarily the fastest. For this throw, the shallowest angle that you can reach home plate with will get the ball there the fastest.
Professor of Aerospace and
Mechanical Sciences
Princeton University
Princeton, N.J.

The SI jump-shot survey quoted in your Nov. 28 article on hot shots included a name that brought back memories of the 1932-33 season. Doug Ash credits John Cooper, who played for the University of Missouri, with originating the jump shot. I agree.

Cooper's jumper was two-handed, both arms about full length over his head. He started with his back to the basket at the free-throw line. After taking a pass, he would hold the ball perhaps 10 seconds or more, feinting, twisting and bending to work the guard behind him off balance. Then he would jump straight up, turn 180 degrees in midair and flip the ball overhead. From squarely in front of the basket it was a high percentage shot. Cooper was known to the Mizzou fans in those days as "Jump Turn Johnny."
Norfolk, Va.

Your account of the Nebraska-Oklahoma contest (What a Way to Wind It Up, Dec. 5) contained an error that stabs deeply into Cornhusker hearts. It grieves us enough that our football team has not beaten the Sooners since the 1971 classic, but for SI to claim that under Coach Barry Switzer "Oklahoma has won by 14 points or more each time" the two teams have met is too much. Let me remind you that just last year Oklahoma had to score a last-minute touchdown to win 20-17.
Lincoln, Neb.

The Fiesta Bowl is indeed ecstatic over landing Perm State. But the Nittany Lions are not our first Top Ten. 10-1 team as Pat Putnam states in his account of the Penn State-Pitt game.

In its six-year history the Fiesta has twice fielded 10-1 Arizona State teams (in 1971 and 1973), and in 1975 it topped all bowls in combined won-loss record when 11-0 and seventh-ranked Arizona State met 10-1 and fifth-ranked Nebraska. That's why we're not shy about calling the Fiesta "America's Fastest Growing Bowl Game."
Executive Director
Fiesta Bowl
Phoenix, Ariz.

Lane Stewart's photograph of Bobby Halpern walking away from a KO'd Diego Roberson (Making a Comeback from Nowhere. Dec. 5) is SI's best boxing shot since Neil Leifer's photo of Muhammad Ali snarling at a supine Sonny Liston in Lewiston, Maine (No Phantom Punch, June 7, 1965). Both are pure art.

Address editorial mail to SPORTS ILLUSTRATED, Time & Life Building, Rockefeller Center, New York, New York, 10020.