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General >> Here Be Maths... >> Formula for Optimum projectile-weight
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Message started by Albion Slinger on Feb 24th, 2021 at 3:10pm

Title: Formula for Optimum projectile-weight
Post by Albion Slinger on Feb 24th, 2021 at 3:10pm
Partly inspired by the 'Ideal Projectile for warfare' thread, I thought it might be a good idea to establish a formula for the most efficient use of projectile mass.

Here's the simple, basic idea:
Kinetic energy / Mass of projectile
= # Projectile Efficiency Units

The higher this number, the greater the ratio of Kinetic Energy to projectile mass.
In order to give momentum and aerodynamics some representation in the formula, I think it necessary for the kinetic energy to be comprised from a velocity average taken over, say, 50y (distances should vary depending on intended range for use)

It might be that the projectile with the highest P.E.U is also the one that flies the furthest, (as long the material and shape stay consistent over different weights).

I'm far from being mathematically savvy, so I'd like to hear some thoughts/criticisms to help refine the idea.

Title: Re: Formula for Optimum projectile-weight
Post by Oxnate on Feb 24th, 2021 at 4:08pm
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.

Title: Re: Formula for Optimum projectile-weight
Post by KnollSlinger on Jun 25th, 2021 at 1:15am
The photon would be the ideal projectile.  That would have very high energy to momentum ratio.

I think the weight of a baseball is ideal.  But make it an
ovoid of titanium.

Title: Re: Formula for Optimum projectile-weight
Post by JudoP on Jun 25th, 2021 at 5:44am
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.

Title: Re: Formula for Optimum projectile-weight
Post by Hirtius on Oct 18th, 2021 at 4:52pm
I think we’re discounting that most identified sling projectiles produced by people are on the lighter side. What was considered the best for warfare might not actually line up with what is most efficient.

Title: Re: Formula for Optimum projectile-weight
Post by 2sloth on Jan 8th, 2023 at 6:42am
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/?utm_source=share&utm_medium=android_app&utm_name=androidcss&utm_term=1&utm_content=share_button

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-0SXT7rjjIs/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.

[/olist]

RDT_20230108_1138456208984649932697399.jpg (19 KB | 8 )

Title: Re: Formula for Optimum projectile-weight
Post by IronGoober on Jan 8th, 2023 at 2:32pm
I'm curious what you used to calculate range? did you include drag and ballistic coefficient, etc? or just parabolic trajectory?

Title: Re: Formula for Optimum projectile-weight
Post by Blindsquirrel on Jan 8th, 2023 at 6:09pm
.

Title: Re: Formula for Optimum projectile-weight
Post by 2sloth on Jan 9th, 2023 at 5:46am
I'm using my own spreadsheet that takes into account gravity, drag and wind (I assumed no wind for this).

Please see reddit post for details:https://www.reddit.com/r/Slinging/comments/10354vr/i_crunched_some_slinging_data/

I'm assuming 10 degree launch angle, drag coefficient of 0.8 (rather high maybe, tumbling cylinder), stone size is based on an assumed density of 2700 kg/m³ (granite) and are assumed spherical with no spin (magnus effect).
Thus the "range" is more of a number to use as an indicator for comparison than absolute ranges.
I might make a goolge sheets version for people to have a go at (and let me know where I'm wrong).
I have recently found some balistic gel penetrations numbers for slow (not from a gun) spherical projectiles, so might include a "penetration" metric to better compare how "dangerous" a projectile might be at range.

Title: Re: Formula for Optimum projectile-weight
Post by Archaic Arms on Jan 16th, 2023 at 5:43am
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.

Title: Re: Formula for Optimum projectile-weight
Post by 2sloth on Feb 1st, 2023 at 9:32am
I have spent some time nerding my way though this and have come up with this spreadsheet for your slinging data:

https://docs.google.com/spreadsheets/d/1jS-0xIEmYL6Bm1J0Jrl7xE_GGeE9rQ9MItkN9Z1qeeM/edit?usp=sharing

This sheet can be used by anyone and has a custom function than can calculate trajectories, kinetic energy, momentum, magnus effct etc.
I also include a "penetration" metric, based on ballistic gel penetration.

So to answer your question for the thread I have two proposal for the "ideal projectile mass" understood as the projectile that is most "dangerous".

I have two suggestions:
1. Range & kinetic energy (at impact): optimal weight 70 g.
2. Range, kinetic energy, penetration & momentum (all at impact): optimal weight 200 g.


Note that both the above are sensitive to the density of your projectiles (I've assumed stone).


sling_danger.png (20 KB | 5 )

Title: Re: Formula for Optimum projectile-weight
Post by Archaic Arms on Feb 7th, 2023 at 7:26am
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!

Title: Re: Formula for Optimum projectile-weight
Post by Blindsquirrel on Feb 7th, 2023 at 7:51am
?

Title: Re: Formula for Optimum projectile-weight
Post by Foresight17 on Apr 21st, 2023 at 2:53am
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.

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