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.

How Lethal is a Projectile?

Rebasing the Scales

In How strong is an Elephant? I talked about how height, weight, and power scale in relation to each other. Explore is a game that uses Logarithmic Scales which double every +n categories. Due to the Elephant post I've revised the scale so that weight doubles every +3, height doubles every +8, and power doubles every +4 (instead of power scaling with weight):

Weight (lbs)	Power	Height (ft)
4	4	4
5	5	4.5
6	6	5
8	7	5.5
10	8	6
12	10	6.5
16	12	7
20	14	7.5
24	16	8
32	20	9
40	24	10
48	28	11
64	32	12
80	40	13
96	48	14
128	56	15
160	64	16

That is, when resizing creatures, moving one category up/down in weight matches to moving up/down one category in height and power, and giving +1/-1 in Strength, Constitution, Damage.

This means power is weight ^ (3/4), and height is weight ^ (3/8). This is as per the Elephant post, but 8/3 = 2.67 which is halfway between my 2.61 value and the 2.79 value in the paper I cited.

Power does not have units as it is applied in different situations. For example it would be the weight in pounds of missiles thrown.

Thus for example a Giant is double the height of a human, so +8 categories. Hence they are approx six times as heavy, and they are four times as powerful.

What affects Lethality

As I discussed in On Throwing, no matter the size of the humanoid, the starting assumption is that they can throw things the same distance - it's just the size of the projectile changes. Hence if a human throws a 6lb spear, a giant is size +8, so throws a 24lb spear with +8 damage due to the increase in mass.

A strong individual gets a bonus on throwing the same spear, because the spear hits the target at a higher speed, because they threw it faster (and hence further). We should choose range so that +1 max range = +1 damage.

Also the lethality of a projectile is obviously affected by its shape.

Hence damage from a projectile is based upon the velocity, mass, and shape of the projectile.

Kinetic Energy and Size

In Sizing Things Up I said that damage of projectiles (given the same shape) was based upon kinetic energy, 1/2 * m * v^2.

Kinetic Energy is clearly the key aspect of the damage potential of a missile, not momentum. If you fire a gun then you experience recoil and and forced backwards with the same momentum as that of the bullet, and this does not hurt you. Most of the kinetic energy is given to the bullet (as it is much smaller) and it is this which makes the bullet lethal.

The problem is that this implies a large wooden ball and a small lead ball of equal weight thrown at the same speed are equally dangerous, when clearly they are not.

(I tried to resolve this problem a couple of years ago but did not arrive at a satisfactory conclusion).

It is clear that the small ball is more lethal because the impact area is smaller. There are two obvious possibilities - divide the kinetic energy by the impact area, or by the square root of the impact area (scaling as the perimeter of the impact area, or the diameter of the cross section).

Dividing by the area is clearly wrong, as in this system being hit by several missiles at the same time would be identical to being hit by a single missile. (5* missiles means 5* KE but also 5*Impact area. Thus KE/Impact area remains the same).

Hence we should assume that we divide by the square root of the impact area, i.e. the damage is proportional to the KE divided by the diameter of the projectile.

Effect of projectile shape

I previously attempted to address this issue, but failed, and kept damage as being KE + a bonus for shape. The difficulty I had before was combining both blunt and sharp projectiles into the same system. Does it make sense to talk of the impact area of a sharp spear, or a round ball? Would they scale in the same fashion as each other?

I recently returned to this problem, and considered - if damage from projectiles is mostly due to penetration - how much energy is required to do this?

Now for blunt projectiles the energy required for this is proportional to the circumference of the hole, as this is how much skin has to be torn, which is proportional to the diameter of the projectile.

My recent observation is that arrow heads rip a hole, the length of which is the width of the arrow head, hence the energy required is also proportional to the diameter of the projectile.

Hence I am now happy that the diameter of the projectile is a reasonable value to divide by, regardless of the type of missile.  That is, a spear and a stone may have different damage bonuses, but it is reasonable to apply the same modifiers to scaling both of them. 

Hence the damage can be implemented as (KE / cube root of size) plus a fixed bonus for the shape.

Range

For a given shape of projectile:

Doubling the range: doubles the kinetic energy (same size), so doubles the deadliness.

Size +12 gives you +12 damage, and you have eight times the power, which means you throw a projectile eight times the mass the same range. That's eight times the kinetic energy, but the missile is twice as wide, so it is only four times the deadliness.

Hence +6 damage = double deadliness = double range or double density. Thus the scale for range and density both double every +6.

Note that for simplicity I'm now assuming that all weapons scale isometrically, so all dimensions of weapons double every +12. This means that giants have shorter swords with respect to their height, whilst halflings have longer swords relative to their height - this doesn't seem unreasonable.

Density

For a given shape of projectile, if you double the density and double the weight (+4), then you've doubled the kinetic energy but kept the same size. So that's identical to double the range (+6).

Hence density doubles every +2!

The New Combined Scale

Combing the results above we have extra columns in our table for Range & Density. For simplicity of figures I'm making Range 10* the numbers as it only applies to large quantities.

Density (lb/sq ft)	Weight (lbs)	Weapon Weight (lb)	Range (ft)	Height (ft)
4	4	4	40	4
6	5	5	45	4.5
8	6	6	50	5
12	8	7	55	5.5
16	10	8	60	6
24	12	10	70	6.5
32	16	12	80	7
48	20	14	90	7.5
64	24	16	100	8
96	32	20	110	9
128	40	24	120	10
192	48	28	140	11
256	64	32	160	12
384	80	40	180	13
512	96	48	200	14
768	128	56	220	15
1024	160	64	240	16

As you can see, a Giant is +8 size (twice as tall, approx six times as heavy), throws a spear +8 size (four times as heavy = 24lb), and gets +8 damage.

Consider a human with a magical throwing gauntlet which gives +8 range, so +8 damage. That's approx 2.5 times as far.

So the four times mass of the giant spear (4*KE) has the same damage potential as 2.5 times range (2.5*KE) because a large spear has a larger impact area (despite being just as sharp and pointy).

Apologies for the maths heavy analysis - but it all leads to a simple table in the rules. Next post I'll discuss complexity and simplicity in rules.