Well, In classical mechanics, kinetic energy (KE) is equal to half of an object's mass (1/2*m) multiplied by the velocity squared. For example, if a an object with a mass of 10 kg (m = 10 kg) is moving at a velocity of 5 meters per second (v = 5 m/s), the kinetic energy is equal to 125 Joules, or (1/2 * 10 kg) * 5 m/s2.

If you then divide by the mass, you get 1/2 x V^2.  So one half of the square of the velocity.

I understand what you're trying to get at but I can't think of a formula to get there right now.

To add onto what Oxnate mentioned, what you are basically doing with this measure is maximising for speed (or half v^2 to be precise)

This leads to the natural conclusion of the lighter the better in all cases (assuming the same amount of energy is put each projectile). Makes sense if you think about it, a lighter object will always have more energy per mass simply because it has less mass but (assumed) the same energy.

Of course things get very complicated when human biomechanics enter the picture. Would you sling lighter and heavier projectiles with the same energy? Probably not... it depends on your technique and perhaps some other factors. The energy put into the projectile is pretty much how much force you apply against the sling cord multiplied by the distance you move against the force applied back by the sling cord (the 'pull' outward as you throw). No doubt there is some sling-technique-ammo setups which allow you to maximise this better than others.

The difficult thing is we are looking at not just the sling, but the interaction of the sling with a very complicated mechanism which is basically human throwing.

...And all that's before you get into the complications of how to assess the destructive power of various ammunition types, another very multifaceted problem with no simple answer.

I don't think there is an analytical formula, but we can estimate for a given sling (and slinger).
I did this for r/slinging on Reddit (data from archaic arms' video): https://www.reddit.com/r/Slinging/comments/10354vr/i_crunched_some_slinging_data...

In summary the optimal weight (most kinetic energy and range) was 50-60 g for this slinger with the longest sling ~1 m

I would guess that for really light projectiles the air resistance of the sling itself becomes limiting for speed.

See Reddit post for details on the plot.
But generally the x axis is centripetal force, and I'm using two vertical axes.

I have made a google sheets version for everyone to have a play at: https://docs.google.com/spreadsheets/d/17iZ7dSNwfxATk-jnWRxRHRe2W-PXTfzb-0SXT7rj
jIs/edit?usp=sharing
I suggest 2 criteria for "optimum" projectile weight:
1. Range & Kinetic energy at range.
2. Range, kinetic energy, momentum & penetration at range.

.

Interesting graph!
It's nice to see the spread between the different slings.
You can see why light bullets were very popular. 45g of lead is very small, so a slinger could carry a large quantity, yet is still highly dangerous when slung with long slings.
On a open battlefield, I think slingers would mostly only use long slings anyway.

This is very curious.
Funnily enough, that is the conclusion that I had come to before seeing this!
70g would be my chosen bullet weight for good range and decent power, and 200g would be my chosen weight for high power with decent range.
Very interesting to see that it lines up with the graph!
Thank you for the effort you put into making this!

I'm in full agreement on optimum projectile weight "in general", however since the average projectile weights for glandes from the ancients appears to be on the lighter side, I have been messing around with 25-35 gram lead balls and glandes. I highly recommend yall head over to TheSlingshotForum and check out the "Power rangers" section, then check out the hunting section. U will find very low energies, typically around 10 fpe, up to 15fpe, taking down rabbits and even raccoons. That is telling me that 15 fpe is atleast very dangerous. With that in mind, I have chronograph readings for the last few days a short sling (about 25 in. From pouch to finger loop) launching 30 gram lead balls and have been getting 150+fps quite easily, and that's giving us about 20 fpe consistently on the low end, 25 on the high end. Make that into a glande and now it has penetration too, on top of being much cheaper and getting to carry way more than 60 gram glandes. I even did a range test between lead balls weighing 60 grams and 30 grams, the results is that the 30 gram balls went on average 192 meters, whereas the 60 gram balls went 170 meters. I think there are definite advantages to light ammo, specifically for warfare.