Hi Joe,

That's a great question... and you've actually made my job a whole lot easier by allowing for all those idealizations, such as ignoring drag, geometry, composition, etc. If you do assume that the singular difference between the two arrows is their mass (and their velocities, of course), then the physics of the situation are greatly simplified.

The two physical concepts that come into play are Kinetic Energy (KE) and Momentum (p). Kinetic energy is usually defined to be:

KE = ½mv²

Using this formula, we can see that the KE of the lighter, but faster, arrow will likely be higher (whether or not it's actually higher, and by just how much, depends on the actual masses and velocities, of course). The best way to go about this would probably be to compute the amount of KE for each arrow (this will depend on their individual masses and velocities). Then, just compare the two numbers (in units of Joules, for example) and whichever is greater will likely penetrate further.

But perhaps the situation might not be quite so simple as all that when it comes to arrows. Normally, the KE can be used to do Work (W = F · d) and this leads straightforwardly to the degree of penetration; this site talks about how the ballistic penetration of armor depends almost entirely on the initial kinetic energy of the projectiles. Poulter Labs also does considerable research into "stress wave and shock wave propagation in solids, liquids, gases, and mixtures".

However, after doing some preliminary research into arrow penetrations, I found a few links that seem to suggest that the tips play a considerable role in determining which kind of arrow penetrates the deepest. Although you did suggest assuming both arrows are identical in terms of composition, tip geometry, etc., I think it might be illuminative to also keep in mind that these other factors might contribute to penetration ability. In case you do consider some of these differences, then the momentum of the arrows (where the momentum p = m · v) seems to gain prominence, as alluded to in the following links:

• http://www.alaskafrontierarchery.com/Articles.html
• http://www.buckdeer.com/kinetic.htm
• http://www.crashbox.com/andrew/harn/digest/year99/99_12_05.txt