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.