Lethality of Guns

My last post was about scaling projectile damage – how to relate changes in size, density, and velocity. As Explore is a fantasy RPG the focus was on low speed projectiles such as arrows and sling stones, but Leland commented about bullets, which I hadn’t considered (as they're not part of Explore), so after some further discussions via email I’ve been digging into the behaviour of bullets.

We discussed several bullets Leland had gathered data for, and in particular compared the slowest, a .32 S&W, and the fastest, a .17 Rem. Whether you measured KE, momentum, or KE/radius, the .17 Rem always came out as much better, which contradicted the opinion on the actual effectiveness of the .17 Rem. (Not any personal experience of mine I should add - I'm vegetarian and have never held a gun!)

Firstly I should clarify something about my formula – KE/radius. It’s a compromise between damage done (KE) and depth of penetration (KE/area). Without any KE there is no damage caused, but without any penetration it’s at best a shallow wound.

High Velocities

Does this formula make sense for high velocity bullets? Let’s consider how penetration is affected by high velocities. At high velocities bullets behave like the target body is a fluid, and we can easily model this like you calculate effects of wind resistance, but unfortunately we cannot calculate total penetration in this fashion since if we model penetration of a projectile into a fluid by a drag force proportional to the velocity squared, then the bullet will get slower and slower but never stop.

Looking for solutions to these equations online, the best resource is actually a GURPS page http://panoptesv.com/RPGs/Equipment/Weapons/Projectile_physics.php which combines equations for both low and high speed behaviour to calculate the depth of penetration:

Km/A * ln (1+v^2/C)

where m is bullet mass, A is cross sectional area, v is impact speed, and K and C are constants calculated from the target material (sectional density of the bullet has already been accounted for). The page I referenced suggests 100m/s for C.

When v is small (<100m/s) this is roughly Kmv^2/AC, i.e. proportional to KE/A. This means the penetration formula for Explore holds true for the range of speeds in that game.

When v is larger (>200m/s), although it is still proportional to m/A, it becomes proportional to ln(velocity), so increases in velocity have a reduced effect.

How does this work out when applied to the bullets Leland and I were discussing? We discussed bullets with speeds 200 – 1200 m/s. That sixfold increase in velocity with the simple method from Explore gives 36 times more penetration, so +32 damage. In the new system for high velocity projectiles it’s only 3 times more deadly, and gives only +10 damage. This means that now the .17 Rem comes out slightly worse than the .32 S&W, as expected.

(Note that since the ineffectiveness of the high velocity bullet was due to friction, I'd rule that the excess energy is used up as heat, so is lost to damage. That is, lethality is penetration * radius, not KE/radius).

Importance of Radius on Damage

It looks like my formula implies that increasing the radius of a projectile reduces its lethality, but actually the reverse is the case; we can rebase it to be density * length * radius * velocity squared. That is, increasing radius increases lethality (it increases in mass more than it increases in radius).

Not however that if you alter radius and length to keep the same mass, my formula implies that long thin bullets are more lethal than short stubby ones of the same mass.
However, there is an opinion that expanding bullets are more deadly, so it was suggested that radius should play a more important role.

Expanding bullets were introduced in 1897 (under the name Dum Dum bullets) as the then new high velocity low calibre bullets were proving ineffective. It was argued by Germany that they were inhumane and they got them banned, but this was based upon poor evidence. Expanding Bullets are actually used today by police forces as the expansion of the bullet slows it down and prevents it exiting the target and injuring bystanders. It seems reasonable to assume that the problem with the old bullets was simply that they went through the target (particularly an arm or leg) and straight out the other side. If you missed a vital organ then the shot was not as effective as it should be. That is, I think the purpose of the Expanding bullet is purely to avoid over-penetration. 

Thus an Expanding Bullet would be ineffective against opponents with tough armour / hide (as it would expand upon hitting the armour and fail to penetrate it) or large opponents, but it might be important in other situations to avoid over penetration and consequent loss of damage. So I'm happy with the role of radius in the formula, but there's a place for situational modifiers.

Deflection Of Bullets

A bullet can also fragment, be deflected, or tumble. All of these appear to have a similar effect to expansion of the bullet - reducing penetration thus giving a shallower but larger wound. The angle of deflection in a collision is determined by the angle of collision and the ratio of the masses of the two objects, so it would appear that less massive bullets are more easily deflected, so this could be a further cause of lower penetration amongst lighter bullets.

Further, for the same reasons, it appears that a lightweight bullet hitting off-centre, hence hitting the armour / hide at an angle, is more prone to ricochet. This would be another reason to avoid lightweight bullets.

Wind Resistance

As a side note it should be mentioned that a faster lightweight bullet with the same KE as another bullet is more accurate over short distance (due to the reduced drop) but over long distance it is affected more by drag, and not only slows down but can be blown off course.

Grazes

Leland also commented that he’d like any system with guns to include a chance of minor wounds or grazes even from a very powerful gun whereas in Explore you can have a sufficient bonus that any hit against a particular target is automatic death. Given the lethality of hits from guns, I agree that you’d have to include something like this, as it changes the dynamic of the game quite considerably.

So, in summary, you could probably have quite realistic rules for guns, and mostly they would simply be in the form of realistic bonuses or situational modifiers for the weapons (as they can be quite simply calculated), but it would take a bit of work. You might want to separate out penetration to be a separate bonus.