## Thursday, October 15, 2009

### Newtons

No matter how many times I sit down to redesign siege weaponry, I am always unsatisfied. I was trying to run a mock combat based on rules I designed before the creation of this blog, and as always, they are cumbersome and unwieldy. So I’ve been thinking about it the past few days.

To begin with, damage.

Let’s begin with an arrow, which we’ll suppose weighs .04536 kg and leaves a bow at a speed of 23.5 m/s. To calculate the force the arrow represents, we multiply kg times m/s over seconds squared, giving us a force of slightly under 1.07 newtons.

This is a force somewhat equivalent to swinging a 1.26 kg dagger (the blade point can be estimated to travel at 0.844 m/s, at least from what I can find online). Yes, an arrow does more damage than a dagger, but one can argue that the force from the arrow is concentrated on its tip, whereas the dagger’s force is spread over a wider surface.

Here’s where my weak knowledge of physics fails me. Alas, I have no calculation to determine the effects of surface area on the force inflicted by an object when it hits. If anyone out there wants to check and challenge my math, I’ll say up front that I am probably wrong somewhere. But I won’t let that worry me and I’ll just keep going.

From what I can find, a large ballista fires a 0.325 kg bolt at a velocity of 65.48 m/s, giving us a force of 21.28 newtons.

Considerably more force than the arrow. If a successful hit from an arrow results in a ‘glancing blow’ off an opponent, we judge that the arrow has caused 1 hp. The above figures would tend to suggest that a ballista bolt which glances off an opponent ought to cause somewhere in the neighborhood of 20 times as much (21.28/1.07).

Yes, I know that the ballista arrow loses momentum faster than the flight arrow, but again, weak physics. I can't find those calculations. If anyone wants to offer, I'm open to looking over your math.

It suggests that damage from siege weapons is severely under-rated, and I think they are. The damage quoted in the DMG for a ballista is a mere 2-12 (3-18 vs. large creatures, a rule I don’t use). Sounds plenty pathetic to me. Minimum damage ought to be 8-32 ... which averages 20.

Suppose instead that for a ballista missile’s damage, first a ‘siege’ damage (SD) is rolled: 1-6, just like an arrow. From this total, subtract 1. The remainder is multiplied by 20 and then 8d4 are added. Thus, if the ballista hits for ‘4’ points of siege damage, the total caused to a living being would be 60 + 8d4, or 68-92 damage. The benefits of calculating it this way will become evident later one.

All right, let’s take a catapult. The best stats I can find (that I can remotely understand from the gobbledy gook that make up these sites) are for a 10.86 kg object fired at 9.79 m/s, indicating that a catapult (onager, mangonel, what you will) has a considerably lower initial velocity in exchange for a much larger missile. The total equals 106.32 newtons of force, almost one hundred times an arrow.

Following the logic thus far, the minimum damage from a catapult ought to be something like 18d6. As before, we could roll an initial SD of 1-6, adjusting the total damage accordingly. Probably, any hit against an actual human person will result in automatic death. The same can be said for hits by trebuchet.

The real effectiveness of the siege weapons is not how large a rock do they throw, but how fast do they throw them. I for one will have to reduce the size of catapult/trebuchet shots as they appear on my equipment table, downwards.

I’ll leave off this for the present, to see if I get a storm from people smarter than me. Tomorrow, I’ll take up one of two problems I see: 1) how often does a siege missile hit more than one person; and 2) what does it do against fortifications.

Incidentally, I know I’m often full of myself, and I know that many who read this blog show great self-restraint in not pointing it out daily. I’m kind of glad for that, but there is not need, really. I don’t bite.

Zzarchov said...

one issue to consider is that force isn't always damage.

You can have a weapon put a clean hole straight through someones arm and do little tissue damage,

while a slower, softer projectile will splinter into the wound and cause more "Damage"

the m16 vs ak47 debate springs up again.

chris said...

First I'd like to point out that newtons are (kg * m)/s^2 and are dependant on acceleration rather than velocity. If two equal-mass equal-velocity projectiles hit something, one comes to a stop after 1 second, and the other comes to a stop after two seconds, then the 1-second projectile has applied twice as many newtons of force to it's target.

http://physics.info/projectiles/

Initial velocity and angle of release determine time of flight (range). Initial velocity and mass determine initial energy. Final energy depends on where in the flight path the impact occurs. If that point is at the same height as release, then the energy is the same (minus drag). Launching from hills or battlements is more effective due to both the increased range and increased energy available at impact.

Wounds are a product of three factors: penetration, energy transfer, and psychological factors. In the case of D&D, psychological factors are taken into account already (in the form of hit points). This leaves two factors to consider.

Penetration is pretty obvious. An arrow will penetrate and make a bleeding hole much better than a sling bullet. Fragmentation further improves the direct damage caused. Most of the available damage tables are based on penetration alone (hence a dagger does less damage than a short sword) except for blunt objects, which are simply balanced to fit the scale of bladed weapon damage.

Energy transfer is less obvious. In Zzarchov's example of the arrow, the projectile enters and exits, taking some portion of it's energy with it. The damage done is almost entirely due to penetration. Contrast this to the case where a catapult shot crushes a person against a sturdy wall. The shot isn't going to go through the person because the force is (probably) too spread out to overcome the tensile strength of their flesh. Some of the force is transferred, the target is thrown back, and moderate damage is done due to compression. Once the target is pinned to the wall, however, nearly all of the energy is channelled from the shot through the target and into the wall, crushing the person into unrecognizable pulp. If it had been only a glancing blow, a modest amount of energy would have been transferred to the target and knocked them out of the way. The shot would continue on with most of it's energy intact, ready to mow down more people. I would guess that your ~100N shot might only transfer 10-30N to a soft target. Hard numbers seem to be very hard to find regarding this sort of impact.

The middle-ground example is a bullet. An efficient bullet stops inside the body of the target, so that all of its energy is transferred. That still leaves a wound track length of 4"-16", but a major portion of the damage is due to the temporary wound track diameter, which can be quite high. Some handgun rounds will open a temporary track 4" across. This will collapse back to a fairly narrow channel, but significant damage is done by that 4" displacement. Compare that to a hydroshock shotgun round that can make temporary track diameters in excess of 12". It doesn't have much more kinetic energy than the handgun round, but the energy transfer is efficient and very rapid.

You're better at bringing all this together than I am, so I'd say have a look at piercing vs. slashing vs. bashing weapons to give you some points of reference for penetration vs. energy transfer. I've looked for energy transfer resources online and I'm not finding much. The actual rates of transfer and the time scales are critical to an accurate analysis of damage potential. If there is a university near you, talk to their physics department. It is a good bet that someone very good at kinetics would be willing to work out the actual energies involved. Even if you can't find a professor, check with people that tutor physics. You're bound to find somebody who loves siege weapons and knows how to determine their energies.

Big McStrongmuscle said...

Small comment: Arrows work fine for the ballista, but you should probably use a sling stone as your model for the catapult. A hit from a blunt object is considerably different from piercing damage, especially against something large like a wall, a giant, or a fortification.

Some quick research suggests that the velocity of a 52-gram bullet leaving the sling is 97.8km/h, or 26.94 meters per second. That's 18.9 Joules of kinetic energy in a sling stone (ignoring wind resistance). Your theoretical mangonel or whatever launches its projectile with 520.4 Joules of kinetic energy. Taking average damage as 2.5 for a sling stone, and going by your hit points/mass guideline, the average damage for being hit by a catapult stone should be ~68.8 points - just about 20d6.