Sling simulation results.
After a lot of testing, and seeing people misunderstand slings badly enough to build ones several metres long, I wanted to actually understand the mechanism. I think I'm a bit closer now, after simulating thousands of paths and visualising the data each time. Two things came out clearly.
A sling loads energy most effectively (fairly intuitively) when the cord is lined up with the direction the hand is travelling i.e. the weight trailing directly behind the hand.
That means letting the sling fly straight out sideways is a mistake — it drops you out of the loading phase early and wastes the release mechanism below.
The trick to keep loading without the weight swinging out ahead is a logarithmically collapsing spiral (the amount varies by length of rope/arm, not speed if constant): the cord stays trailing because the path tightens faster than the weight can overtake it. You can see this in a lot of techniques as a final overhead winding motion, which matters more than it looks.
Eventually you can't feed or tighten the spiral any further — by hand that's basically the single meaningful whip right before launch. You throw it out, and the cord releases the energy stored in that trailing position, swinging round to its maximum speed when it's perpendicular (90°) to your hand's motion. Trying to keep driving it during this last bit is much less efficient.
So the sling's real output is a rotation that's perfectly in phase with the hand: the stored spiral energy plus the hand's own velocity, adding together. In phase, my testing says you can get ~70% more speed than a rigid arm of the same reach.
On length, it comes out exactly 1:1 with the arm. During loading, the arm and cord form a right triangle from your hand, and the energy you can store scales with the area of that triangle — which is largest when the two sides are equal. Same reason a square has more area than any rectangle of the same perimeter**.** So making the cord longer doesn't just make the spiral harder to drive — it also hurts the 90° release energy. Longer is not better. There could be also other strange geometric reasons that contribute to why that ratio is more efficient, or the others are a lot worse, but extending the cord is always negative.
A bit messy — the blue dots are just how I guide the curve. The white line is the hand; you can see it stays perfectly trailed at every point as the spiral collapses faster than the weight can overtake it, then flicks back out for peak ball speed at the 90° correction.
Below is the visualized snap.
Hopefully the findings are useful to someone / help mentally clarify some technique. There may be some mistakes but I am fairly confident.