Well, it works like this (sort of):

First, you have to accelerate the stone, in which process your hand always leads the pouch and pulls it around the circle.   The hand moves around a circle of small radius, leading the stone.  The stone move about the same center, on a larger radius, following the hand.  The radial acceleration and velocity are equal or nearly so, but the linear velocity is multiplied many times as the stone moves along its larger circumference in the same amount of time as the hand on its smaller circumference.  It matters not what style or how many circuits the thrower makes, this principal applies for the most part.

During the windup, the sling resistst the stone's inertia and tendency to leave its orbit in a straight line.

Now, there is the matter of aligning the pouch for the release.  This involves altering the plane of the stone's orbit.  The sling and stone behave like a gyroscope, resisting this change, consuming the energy used for the move.  My guess is that most of us do the first two steps slowly until the stone is on the plane we want, then accelerate the stone.  This is because it takes less energy to changes planes when the stone is moving more slowly and the gyroscope effect is less.

Then comes the release.   Upon release, the stone leaves the pouch in a straight line (in theory at least it is only dropping, not curving.)  This means that the sling must be at a right angle to the target, or plane of trajectory, at release.  The stone has to overcome the inertia of the release cord and pull itself out of the pouch.  The means the stone loses some velocity and the slinger must have good timing.

This all takes only a very short time. 1/2 second with a single windup to maybe 2 seconds with multiple windups.  With lighter "stones" I use a modified diagaonal, single windup throw.  The sling makes one curcuit in which the stone moves thru 360 degrees horizontally and 135 degrees vertically.  The stone begins hanging and is released at 45 deg. down from vertical.  With heavy stones, I find the underhand throw more effective for control.  I still use one windup.

The overhand throw I find ineffective in many ways.  Not the least of which is that by virtue of the style, the shoulder becomes a point of high stress.  Like the point of a pole vaulter's pole, the arm, sling, and stone rely on the shoulder to take all the stress and energy.  One has to lever the entire throw with the shoulder.  The mass of the arm added to the sling and stone makes for more difficult acceleration.  And the overhand throw only passes throw about 180 degrees, meaning less time to accelerate.

With the other styles of throw, the stress is distributed on to the other joints.  And it takes much less energy to spin the wrist and hand with the elbow.

Now, the 'kicker':  at release, many of us "flick" our wrist, adding enough last minute acceleration to compensate for the stone's need to overcome the release cord inertia as it pushes its way out of the pouch.  This is sort an afterburner in the throw, adding some punch at takeoff.

Now, I don't have the slightest idea about the math that would describe any of this.  And this is not all that happens  Additionally, if any of the more experience slingers say something different than I have, you misunderstood me.  I agree with them.

jeff <>< - who thinks this is going to be a great thread

Nice work Chris!