Sunday 5 February 2017

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.

5 comments:

  1. I'm home sick today (boo), so I have some time to think about this and respond (um, yea?).

    I've been thinking about this issue myself lately. I have some doubts about KE being the key to lethality (although clearly there's a relationship there), and I think you're rewarding projectiles with smaller diameters/cross-sections too much. It seems to me that what you're really measuring by dividing KE by diameter is penetration ability, which is not quite the same thing as lethality.

    In the case of bullets, the (sadly ill-defined) concept of "stopping power" is generally associated with higher calibers (i.e. larger diameters). This makes some intuitive sense: a larger projectile makes a larger hole and is more likely to damage "important stuff" (apologies for the medical jargon) on the way through the body. And if you consider the limit of an infinitesimally-thin projectile, which would have extremely high penetration (assuming enough KE, naturally), it would create such a small hole that it might not actually do all that much damage unless it went through a very vital spot.

    Also related (and reminding me of what I wanted to say about your post about Trip & Disarm and critical attack rolls back in September of last year): projectiles with high penetration (high-powered rifle bullets, for example) often have lower lethality than one might expect due to their KE because they fail to deposit all of this energy in the target. That is, it is not uncommon for such projectiles to pass completely through the target, especially if they don't strike something hard (typically bone) and fragment within the body. So at least arguably a high-power projectile is not all that much more lethal than a lower-energy projectile, UNLESS its trajectory is such that it passes through a vital part of the target. (For example, a rifle bullet might well penetrate the skull and the brain, while a .22 pistol bullet probably bounces off the skull. But both will probably pass though and make a hole in the meat of the target's arm, say.)

    The concern I have with high "kill" values for attacks is that they can make it impossible to inflict what I call "Captain Kirk" wounds (grazes or scratches, named after how Captain Kirk would have an axe or something swung at him and end up with a ripped shirt and a red line on his chest, but no other injury). Once your "kill" bonus is high enough, even the lowest possible kill roll is guaranteed to wound or even kill the target. The solution I have hit on is to by default limit attack kill bonuses to a value related to the (unarmored) damage thresholds of the target, such that there is always a reasonable chance of a graze/hit in a non-vital spot that is unlikely to inflict serious damage. (Given the open d10 roll system, a lethal result is ALWAYS possible, even if extremely unlikely.) However, for higher attack rolls, every pip by which the roll exceeds the target number/defense value allows one extra bonus to the damage/kill roll, up to the maximum value for the projectile. Thus an attack roll that "barely" hits the target just indicates a hit somewhere within the target silhouette, and you need to rely on a high kill roll to indicate placement against a vital spot, while a very high attack roll represents a well-placed attack allowing the attacker to utilize most or all of the damage potential of the weapon (representing a "center mass" hit for a firearm, for example).

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    1. I did read up a fair bit about lethality of guns, but the websites were nearly as disturbing as those on Archery effectiveness. There seemed to be little agreement about anything! It is true that my system is at best a massive simplification - but as far as I understand it, larger calibre bullets allow for a larger mass to be fired at the same velocity, hence they would be more lethal in my system. For applicability to guns you’d have to list various guns and stats about bullet mass and velocity, calculate the lethality, and see how the results correlate to accepted wisdom.
      Any kill roll can hit but not wound (by rolling a double 1), but that means it still tops out at 99% chance of death from a hit. The "Captain Kirk" effect is partially modelled via him having a high save bonus from his level, so he would quite easily get a minor wound (-1 penalty) from a blow likely lethal to an average human. Your max kill bonus is an interesting idea, but the +1 kill per +1 you hit by is a bit fiddly (hence why I instead give +1d10 for each +10 you hit by). If you want a simpler alternative – how about the maximum wound you can inflict is the max number you rolled on the two dice? Then you’d need to roll a 5 or better to get a kill, so a 16% chance of cannot kill. (A 0 counts as 10). Again you’d have to run the numbers to see if that works out.

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    2. Agree that there's very little scientific data about projectile lethality out there, and a lot of opinion (and disagreement). Re: compiling a list of ammunition and corresponding mass, diameter, and velocity, I've actually done that -- a few years back I found a source that listed a ton of modern and historical cartridges with info on caliber, bullet mass, and muzzle velocity, and set up a spreadsheet with energy, momentum, and so on. Caliber and bullet diameter are not generally related in the way you suggest, though -- often higher caliber rounds have lower muzzle velocity, but are often felt to have better "stopping power." This varies a lot, though.

      Because of some received wisdom that projectile size (specifically, diameter, and to a lesser extent mass) was more important than velocity, I scaled my results using momentum. But thinking about it now, this idea is probably driven by the fact that high-velocity projectiles are less likely to deposit all of their energy in the target. (Well, plus the effect of the larger hole.) Maybe I should combine my limited lethality unless the aim is very good with a KE-based scale. I agree it's a bit fiddly, I need to think about it some more. I've just starting doing some occasional playtesting to see how it feels.

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    3. I think you meant to say caliber and muzzle velocity are not related, but my point was that larger caliber does not mean necessarily lower muzzle velocity. That is, the division by the radius doesn't mean larger bullets are less lethal in the above system. I would be extremely sceptical of received wisdom in this subject!

      If you find that spreadsheet I'd love to have a copy. PM me it to Joe on the www.therpgsite.com forum.

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  2. Yes, you're right, I meant caliber & velocity. I know what you mean about received wisdom, but sometimes it's the only wisdom available. I'm reminded of the book The Physics of Baseball by Robert Adair in which he said something to the effect that we shouldn't ignore (for example) professional baseball players when they say that doing something in a particular way helps them. They may not be correct about WHY it helps their performance, but if they feel that it does, it's at least worth consideration. (Not that they should be believed in all things, either...)

    I have the spreadsheet, and I see that I cleverly referenced the source, which is here: http://www.aosurplus.com/ammo_chart.htm. It doesn't appear that therpgsite.com supports file attachments in PMs so I'm not sure how to send my spreadsheet to you. If you want to email me directly so I can reply with it in email, drop me a line at leland53 at comcast dot net.

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