The third equation shows the change in range due to an error in launch velocity (?v)
?R= 2v ?v sin2?/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 2?=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 ? = 45� (cos 2? = 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 ?v 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.
ENOCH J. DURBIN
Professor of Aerospace and
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."
RICHARD C. MONTAGUE