Sectional Density

[mcb]

There is a linear relationship, assuming the target medium is homogenous and the comparison is between bullets of similar frontal area that do not tumble.

Should be a square root of the velocity for non-expanding projectiles in a non-fluid based homogenous target. Penetration is proportional to kinetic energy. This was shown going back to the late 1700 when physicists where still arguing the definition of what the energy of motion is. It was shown that steel balls dropped into clay had their penetration proportional to the square root of the velocity. You can see the relationship between kinetic energy and penetration in the armor penetration tables for non-energetic based AP rounds for everything from small arms to tanks to battleships. Somewhere I have the table for the 8.8cm kwk 43 (main gun for the Tiger II and a few other vehicles of WWII) that include not only the penetration at range but the measured impact velocity's, the the relationship holds very well.

 

[JohnKSa]

For an impact where there is little or no "splash" effect in the target medium, that's exactly correct. Kinetic energy is related to mass displaced--that's the definition of kinetic energy--mass moved over a distance. If the mass is displaced by essentially boring a hole that is roughly the diameter of the projectile into a non-elastic material then kinetic energy is going to be related linearly to penetration.

If there is significant mass being displaced other than just boring a hole, then energy is being used to generate that displacement and therefore less of it is available to drive the projectile deeper. Where there's a lot of "splash"/radial stretching effect at bullet impact, displacing the splashed/stretching material uses up energy which is then no longer available to create penetration. In that situation, impact momentum can give you a better idea of penetration.

You can see the relationship between kinetic energy and penetration in the armor penetration tables for non-energetic based AP rounds for everything from small arms to tanks to battleships.

I think that makes sense as non-energetic AP is about essentially boring a hole that is roughly projectile diameter.