Addendum: In the face of Professor Pi's extraordinary mathematics, I have been forced to reconsider my write-up. I think Ben Franklin said it best when he said that the greatest tragedy in the world is when a beautiful theory is murdered by a gang of brutal facts.

This node has earned me a nice chunk of XP though, and reworked it may yet prove feasible (there is a little bit of historical data that wasn't made completely impotent). So for posterity's sake I'll leave my unedited version here. Besides, what good is a counterpoint without its point?

The genealogy of the granny shot is lost to me, and hours of tedious research have done little to alleviate this situation. It was likely developed by a rather athletic individual before the evolution of basketball into a national passtime.

The granny shot is an underhand throw of a basketball at the hoop. The name was derived from the resemblance of the shooter to a hunched-over old hag. It is often considered that it should only be used by children and special people. However, contrary to popular belief, the granny shot is scientifically the superior way to shoot a free throw. There are several reasons for this:

  • Trajectory: A standard NBA-approved basketball has a circumfrence between 29 to 30 inches (for the non-prodigy or calculator-deprived, that is a diameter of 9.40 inches). The hoop itself has a diameter of 18 inches. In other words, one circle is quite a bit bigger than the other. Now, imagine the trajectory of the standard free throw. It starts at the player's hand and goes in a low-curve arc towards the basket. Imagine the hoop from the ball's perspective. It is entering at a very slight angle. Instead of aiming for a circle with a 8.60 inch margin for error, it is aiming for an ellipse with a much narrower margin. A granny shot, however, goes up and down at an arc with a much higher curve. It goes more down than forward, making our basketball-cam see more circle than ellipse. The greater margin consistantly grants greater accuracy on free throws.
    Note: The highest margin of error would be from an extremely high arc. However, with the more power applied, accuracy would suffer, undermining the advantage.

  • Spin: Much like the English on a pool ball, the spin on a basketball affects how it will enter the basket. The common overhead free throw puts a backspin on the ball; the granny shot puts frontspin on it. Imagine the ball hitting the front of the hoop. With backspin, it hits, bounces back, and misses. With frontspin, however, it goes forward--into the basket! If it hits the backboard, the arc of descent will cause it to drop directly into the basket after impact. The arc and English of the free throw will bounce the ball back to you. No point. You've let down the team.
  • Muscle: For anyone who has ever lived a non-sedentary lifestyle, it is common knowledge that the biceps (which flex the arm) are stronger than the triceps (which straighten it). With the normal free throw, the arms are straightened while shooting. It requires more force, which results in less detail and subtlety in those ever-sensitive motor neurons. With the granny shot, you are using your strongest arm muscle. Actually getting the ball to the hoop is not a concern of those pythons. That leaves you to concentrate on more important things, like actually scoring a point.
  • Control: Almost every professional basketball player shoots in the classic style. One hand behind the ball, one hand steadying it, over the head. To represent why this is inefficient, try to balance a golfball on your index finger, while using your other index finger to steady it. Then try to balance it while holding it with both fingers. It is much easier to control the ball when both hands are on it. Control means points.
  • Sight: A lot of things can go wrong while trying to make a basket. Having the ball in your line of sight can really help your shot. You can see the ball from start to finish. Any minor adjustments can be made, resulting in that precious free throw.
The granny shot should not normally be used in regular play, because people will have their arms up and attempting to block your shot. The jump shot gets over these obstacles--the granny shot is a poor idea in this case.

Now, some people may offer the repartee of “Well if it’s so cool, why isn’t EVERYONE doing it?” Well, basketball is a spectator sport, particularly in the United States. The sportsmen are showmen as well. The granny shot is associated with children and old people. It’s a blow to their pride to use it. However, there have been actual NBA players to use the granny shot. As a matter of fact, former record holder Rick Barry, who retired with a .900 free throw percentage using what described as his “odd, outdated underhand style.” There was even a time when he missed only 9 free throws in an entire season. He is ranked 15th top scorer of the NBA and 6th of the ABA. Those are some impressive numbers. It’s all about the physics.

Rist, Curtis. DISCOVER MAGAZINE “The Physics of Foul Shots: Underhanded Achievement” NBA Legends

And I thought this node was about something completely different...

Nevertheless, Trippin brings up an interesting point: why do basketball players generally use the overhand shot, and not the underhand, or granny shot? Since it is apparently all about the physics, let's get our hands dirty and toss in some equations.

Consider an average basketball player: I'll call him Michael Jordan in this example. Mike is a decent 1.98 m tall forward (I'll stick to the metric system, since that makes calculations somewhat easier.) Mike has 0.80 m long arms, extending from his body at a height of 1.69 m. Of course, I just made that up, but let me assure you that MJ's proportion are well in line with the Golden Ratio of the Human Figure. He's just like that naked guy in the soap bubble by Leonardo da Vinci, remember?

Michael can either shoot the overhand free throw, or the underhand. See the following stick figures. In the overhand free throw, Mike releases the ball with his arms stretched out at 45 degrees. In the underhand free throw, he lets go of the ball with straight arms.

     O /           O 
     |/            |    *
     /             ----
     |             |
     |             |
     |             |
     ~~            ~~
   Overhand     Underhand

The horizontal distance from the line to the hoop is 4.57 m. Actually, the shooting distance is a little less, because in the underhand free throw, Mike's arms are extended 0.80 m forwards. In the overhand free throw, Mike's arms are extended 0.57 m forwards, and 0.57 m up (Pythagorean Theorem, you do the math.) The rim is suspended at a height of 3.05 m.

         *                              |
     O /                                |
     |/                                 | 3.05 m.
     /                                  |
     |                                  |
     |                                  |
     |                                  |
                   4.57 m.

As Trippin already mentioned, the entry angle for the ball is crucial. A shallow entry angle (a "flat" shot) has little chance of success, since the apparent opening of the hoop is small. On the other hand, a large entry angle (almost vertical) would require a very high arc, and thus a lot of energy. The ideal entry angle is somewhere around 45 degrees. I won't bore you to death where I found this fact.

Now, if you didn't totally sleep during Physics 101, you would know that the trajectory a projectile (a bullet, a ball, or rotten tomatoes) describes is a parabola. The general equation for a parabola is:

y = Ax2 + Bx + C

The derivative of this function is also important, since it describes the angle the ball will make, as a function of distance from the rim. We can use it to calculate the highest point of the arc, and the entry angle.

y' = 2Ax + B

For the overhand shot, Mike releases the ball 0.57 m. beyond the free throw line, at a height of (1.69 m. + 0.57 m. =) 2.26 m. Hence, y=2.26 at x=0.57. If all goes according to plan, the ball ends up in the hoop, and thus y=3.05 at x=4.57. The ball approaches the rim at a 45 degree entry angle, or a tangent equal to -1. Hence, y'=-1 at x=4.57. So we have three unknowns: A, B, and C, and we can set up three equations to solve for these. Let's skip some of the boring math and give you the answer:

y = -0.2994x2 + 1.7363x + 1.3676

Similarly, for the underhand shot, we have: y=1.69 at x=0.8, y=3.05 at x=4.57, and y'=-1 at x=4.57. This results in:

y = -0.3609x2 + 2.299x + 0.0818

Hey hey! Look at the quadratic (A) coefficients: the underhand shot has a larger negative value, so it has a higher curvature. Just like we expected... However, remember that both shots have the same entry angle of 45 degrees. The entire trajectory of the ball is irrelevant; it only matters how the ball approaches the rim.

Now you can calculate how high Mike has shoot the ball in the overhand and underhand method. But don't strain yourself: I calculated it for you. The overhand method reaches the top of its arc at 3.89 m. The underhand at 3.74 m. That doesn't look so good for the overhand method, but remember: the overhand shot was released at a greater height. From its release, the overhand shot travels 3.89-2.26= 1.63 m. up, while the underhand shot moves 3.74-1.69= 2.05 m up. For an official 0.6 kg game ball, the overhand shot requires 9.6 J, and the underhand shot 12.1 J. The granny shot actually requires 25% more energy!

As Trippin points out, spin is indeed also a factor to consider. However, I claim that backspin is actually a good thing. This is because of the Magnus Effect: the backspin causes the ball to reach its highest point of the arc closer to the basket, and the ball will have a (favorable) steeper entry angle. If the ball lands on the rim, all bets are off -- either with backspin or frontspin. Furthermore, professional basketball players seldom use the backboard for free throws.

Muscles is indeed a matter of biceps versus triceps. Although the biceps are generally somewhat (though not a lot) stronger than the triceps, you actually have a lot more from the latter. About 60% of the arms' muscle-mass is in the triceps. This suggests a better control over movements. Speaking of control: a professional basketball player rolls the ball of his fingers in an overhand shot as well (a.k.a. follow-through.)

And last but not least sight: in the granny shot, you will have to look down to see the ball. In an overhand shot, both the ball and the basket are lined up with the eye. The shooter is looking in the direction of the shot. This is a huge advantage.

As far as coolness is considered, I don't think any coaches care. I'm sure Phil Jackson would have Shaquille O'Neal shoot in granny style if it could improve his laughable stats.

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