From Hooke's Law:
F = -K.x
Elastic energy:
E = (K.x.x/[2.L])
K = elastic modulus
x = extension of band
L = original length
k should be the same for bands of the same material, not related to thickness of the band.
x and L can be measured.
This will give the amount of energy for each system, dont forget to apply this to each band used.
You then need to look at the momentum of the shaft and equate the energy in the system = 1/2 .m.v.v
This should tell you the velocity of the spear leaving the bands.
m is the mass of the spear and will be a function of the length and diameter.
You could use this to get an idea of equivalence between bands and guns, however i dont know if the speed of the shaft leaving the band is directly related to the distance it can travel, but i think so.
I hope someone else can add to or correct this...
The problem with this is that it does not take account of the number & diameter of the bands used -- which I think is what is being asked. It doesn't answer the question.
From earlier threads, it was proposed that the power of bands, for the purpose of comparison, is proportional to their cross-sectional area. Whether this is strictly true or not, I don't know. Also some like to subract the area of the hollow inner. An additional factor to take account of is band length. If you look at the band chart from Rob Allen or Rabitech, you will see that they recommend longer lengths for fatter bands - because they would be too difficult to load otherwise. So their extension will be less. To further confuse matters, spearguns using more powerful rubbers usually come equipped with thicker, heavier spears (I think spears that are too thin can flex when released(?)).
As best I can recall the "power" comparison of the various configs looked something like this:
14mm < 16mm < 18mm < 2x14mm < 20mm < 2x16mm < 2x20mm
So
maybe you could use something like this: F = k x AreaRubbers where k is some constant
Assuming the spears are cylinders of the same material: Mass = density x volume = density x AreaSpear x length = density x length x 3.142 x (diameter/2)(diameter/2)
Density is roughly the same for both spears.
The focus on power sometimes overlooks the fact that we often already have enough power. I'm currently using a single 20mm band (my only option at the time) on my 90cm railgun but will switch back to the original 16mm next time if I can get some - as it was plenty powerful enough with that and is much easier to load. My new 75cm gun comes equipped with 18mm eurobands - haven't tried it yet though.