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Altair_the_Vexed
2009-10-22, 07:02 AM
I've never been entirely satisfied with the falling damage system in D&D in any of its incarnations: it's purely linear, but acceleration due to gravity is a square function.

Now I don't want to kill any catgirls with applying physics too rigidly to D&D, but I do have to make a system that uses some kind reality in the mathematical function.

After much examination of terminal velocity data for big and little creatures, skydivers and the like, and looking at the equations for velocity of a falling object (on Earth), and d20's sensible rules for altering damage dice in collisions based on the smallest object in that collision, I've come up with this:

Falling damage = XdY where X = (square root of height [rounded down]) -1 where Y = die type for your size category

Y starts at d3 for Tiny creatures, and increases by size category.

(Tiny = d3, Small = d4, Medium = d6, Large = d8, Huge = d10, Gargantuan = d12, Colossal = d20)

Fine and diminutive creatures take no damage from falling.

Maximum falling damage is a number of dice equal to the square of your size category.

(Tiny = 3 , Small = 4 [squared to 16], Medium = 5 [squared to 25], Large = 6 [squared to 36], etc)

The rational behind this set of figures?Velocity from falling = square root (height fallen * 64) [feet per second]

[s]dealing 1 die of damage per 10 fps
~ (square root (Height fallen)) -1
Terminal velocity for humans is in the region of 200 - 300 ft/s
Wind resistance to acceleration from low mass / high surface area creatures, such as diminutive and fine creatures like hamsters and spiders is low enough that they never exceed a few feet per second.

Under this rule, characters are subject to less falling damage from great heights, and slightly more from lower heights.

To avoid high level fighter types jumping off 200ft drops and absorbing the damage as a means of ordinary movement, you run the risk of CON damage from any fall over twice your height.


Roll 1dQ, where Q is the die type closest to the number of dice used to roll damage.
Fortitude save (DC = 10 + number of dice) halves.

PEACH...?

After all the helpful physics lessons, I've come to the conclusion that the linear increaee in damage is the most sensible, realistic and simple method.

Falling deals XDY damage to Tiny or larger creatures (and diminutive or smaller objects, but not creatures).

X is number of feet fallen / 5 Y starts at d3 for Tiny creatures, and increases by size category. (Tiny = d3, Small = d4, Medium = d6, Large = d8, Huge = d10, Gargantuan = d12, Colossal = d20)
Maximum damage is 10 x die size - e.g. 60d6 for Medium sized creatures.
For numbers of dice over 10, you may wish to use the average damage.
Grant a Tumble (or Dexterity) check to reduce the damage from the fall.

If the Tumble (or Dexterity) check exceeds DC 5 + number of dice (maximum 50), you take half damage. If the Tumble (or Dexterity) check exceeds 10 + 2 * number of dice, no damage is taken.

Zeta Kai
2009-10-22, 07:23 AM
Well, I can't fault your math (at least, not without some text books handy), but it seems a bit complex for a game rule. I agree that the D&D rules for falling damage are troublingly simple, but this method is probably going too far in the other direction. I suggest some simplification of your rules to enhance ease of use in actual gameplay. After all, no matter how accurate a rule may be, a rule that never gets used is a bad rule.

Regardless, you get an A for effort. :smallcool:

lesser_minion
2009-10-22, 08:12 AM
Actually, linear isn't as bad a model as you might think.

'Damage' essentially amounts to 'work done in injuring you' and depends on two main factors:

The amount of kinetic energy available The amount of kinetic energy that actually goes into injuring you (essentially, the 'damage efficiency')


The second factor is usually taken into account by rolling dice - a low roll means that you landed in a way that minimised the damage done by stopping.

Neglecting drag, you would find that the maximum amount of available kinetic energy for injuring you can be approximated as:

Mass x Acceleration of Free-fall (9.81 ms-2) x Height Change

Which is also the amount of potential energy you would lose in falling.

The energy available to injure you is actually proportional to the square of the velocity with which you hit the ground.

Altair_the_Vexed
2009-10-22, 08:34 AM
The energy available to injure you is actually proportional to the square of the velocity with which you hit the ground.
Absolutely, but as the velocity increases, the energy increases by a squared factor...

I don't think this method is as complex as it looks.
The hardest bit is figuring the root of the height, and as one is rounding down, one only has to work out which square number you have exceeded.
90 ft drop? >81, so 9dY. 150 ft drop? >144, so 12dY.

Changing the damage die is vital, in my opinion. Children take less damage from falls than adults, chiefly because their momentum is less (I work in child safety engineering, I have stats for this). Similarly smaller creatures have lower momentum than larger creatures - and as Lesser _Minion pointed out, damage is due to the energy being changed from your motion.

I noted the maximum damages as the squares of their size categories, but we can easily show that in a table:
{table]Size | Falling Damage Die | Maximum Falling Damage
Tiny|d3|9d3
Small|d4|16d4
Medium|d6|25d6
Large|d8|36d8
Huge|d10|49d10
Gargantuan|d12|64d12
Colossal|d20|81d20[/table]
I chose to avoid the table to show that the figures can be readily worked out from first principles.

Random832
2009-10-22, 08:49 AM
Since people are confused as to just what is a square function of what, I've done some formulas.

Key:
h = height (or distance fallen)
g = acceleration due to gravity [constant, 32 ft/s]
t = time (taken to fall)
v = velocity (at end of fall)
E = kinetic energy at end of fall ( = potential energy at start of fall)
m = mass of falling object



I've never been entirely satisfied with the falling damage system in D&D in any of its incarnations: it's purely linear, but acceleration due to gravity is a square function.

Actually, acceleration due to gravity is a constant.*

Given a constant acceleration, it is position vs time that is a square function. That's the one you're thinking of. But that's not really the most relevant question. It would be more relevant if we were calculating the amount of time that it takes to hit the ground.

velocity vs distance fallen is actually a square root function (i.e. v is proportional to √h )

h = 0.5gt
[v = gt]
[t = v/g]
h = 0.5g(v/g)
h = 0.5v/g
2gh = v
√(2gh) = v

Which makes kinetic energy vs distance fallen a linear function:
[√(2gh) = v]
E = 0.5mv
E = 0.5m√(2gh)
E = 0.5m2gh
E = mgh

So the use of a linear function is actually absolutely correct. It's just that hit points are a crappy abstraction, so the 20th level barbarian walks away every time.

* Technically it is an inverse square function, GM/r where G is an absolute constant, M is the mass of the earth and r is our distance from the center of the earth. But r does not change significantly enough in this context to warrant not treating it as a constant

lesser_minion
2009-10-22, 08:57 AM
The 'damaging efficiency' of a particular fall is probably higher (i.e. more dangerous) for both larger creatures and less agile creatures (which are less capable of moving around in the air), but my argument was that the potential energy change (approximately m x g x deltaz) should determine the number of dice rolled - the type of dice used should depend on how 'efficient' you think the impact is at hurting people.

The same principle gets applied a lot in D&D and its derivatives, although sometimes people forget about the 'efficiency' term (which decreases with velocity for many kinds of projectile, once it reaches a certain limit).

It's the energy that counts - the momentum is a separate issue, and is better used for proving that we do not have bullets that fling people across a room. Try water cannon.

Random832
2009-10-22, 09:02 AM
Of course, none of this takes into account that terminal velocity is not a sudden shift from "accelerating at 32 ft/s" to "not accelerating at all". Anyone know the formula for air resistance?

Another question is how we model 'where the ground is softer'. Reduce damage cap, reduce die size, apply a constant reduction to damage, allow an easier save, save for less than half? Below a certain height, falling into loose sand is going to be less dangerous than falling onto stone, and falling into water* less dangerous still. (This also brings up the separate question of landing on rough terrain)

*Yes, people say that from high enough water is "like concrete", but they're wrong - It's certainly dangerous enough to cause injury, but those are heights from which someone falling onto concrete would be reduced to a paste.

On the proposed die sizes - d20 is a bit of a large gap compared to the others... I'd make the damage die for colossal 2d8.

lesser_minion
2009-10-22, 09:12 AM
Of course, none of this takes into account that terminal velocity is not a sudden shift from "accelerating at 32 ft/s" to "not accelerating at all". Anyone know the formula for air resistance?

The important part for these purposes, as far as I can recall, is that drag can be taken as being proportional to either v or v^2 (which one varies depending on how fast you are going).

As for landing on softer ground - reduce the 'damaging efficiency' again. There would probably be some sort of maximum reduction, but at its most basic, a softer surface would absorb some amount of your energy by deforming, and the energy so absorbed wouldn't damage you.

I don't think something like 'reduce damage by -2 per 10ft. fallen, maximum of something' would be too horrible from the naturalism perspective, and I doubt that it would be a problem to resolve either.

I'll probably be looking at some fluid dynamics some time tomorrow, so I might be able to provide some more useful assistance then.

Random832
2009-10-22, 09:18 AM
So:
The number of dice models "veesquared" (approximated as linear with height)
The cap on the number of dice models terminal velocity
The die size can be changed to represent mass?
The randomness factor represents random variation 'damage efficiency'?
The no damage to nonlethal to lethal progression in RAW represents velocity-related effects to damage efficiency?

lesser_minion
2009-10-22, 09:27 AM
I think that's pretty much it.

The 'damage efficiency' basically takes into account the ability to dissipate some energy 'safely' - this includes energy loss as sound, deformation experienced by whatever you hit, and deformation experienced yourself.

In this case, the faster you slow to a stop, the worse your injuries are likely to be (this works reasonably well as a general rule).

It might be worth explaining that hitpoints model injuries by assuming that the number of serious injuries a character receives equals the sum of the probability each injury they sustain is serious.

Altair_the_Vexed
2009-10-22, 09:51 AM
(Root function / suare function nomenclature confusion: apologies!)

This is getting away from the relatively elegant (I thought):


velocity is proportional energy, which is proportional to damage

which I modelled by increasing number of dice porportionally with the velocity, changing the die type according to mass, and setting caps according to terminal velocities in the real world (based on 10 fs^-1 = 1dY).
I used the velocity vs distance equation to get my root(h) function, after a lot of simplification from the [√(2gh) = v].

Does the drag really reduce the acceleration to a linear increase?
If so, then our velocity increase must be less than the pure 32 fs^-2 that I originally modelled... which means the 1dY per 10 ft drop rule isn't right, either: it's too high.*

*Assuming that 1dY per 10 fts^-1 is good.

lesser_minion
2009-10-22, 10:08 AM
(Root function / suare function nomenclature confusion: apologies!)

Does the drag really reduce the acceleration to a linear increase?
If so, then our velocity increase must be less than the pure 32 fs^-2 that I originally modelled... which means the 1dY per 10 ft drop rule isn't right, either: it's too high.*

*Assuming that 1dY per 10 fts^-1 is good.

The drag experienced by a moving body is directly proportional to its surface area and a function of its velocity (either v or v^2, but I can't remember the criterion for each). Because the drag is a force, however, the amount of effect it has on the acceleration is inversely proportional to the mass of the falling body.

About this point, you'd have to break out some calculus, which is why the designers stuck to just imposing a maximum amount of fall damage.

The integration could be done behind the scenes, in which case you would be able to offer a table and a formula, which would probably be fine for gaming purposes.

My argument was that it was the energy which determines the damage done (and is directly proportional to the square of the velocity with which the object hits the ground).

Translated into rules, the points we made shouldn't actually be as horrible as they sound.

How far the damage someone takes from something depends on force and how far on energy might vary - at its most basic, breaking things always takes energy, but the exact response depends on the forces the falling object is subjected to in stopping.

Random832
2009-10-22, 10:18 AM
(Root function / suare function nomenclature confusion: apologies!)

This is getting away from the relatively elegant (I thought):


velocity is proportional energy, which is proportional to damage

Well, right - the problem is that velocity isn't proportional to energy - the square of velocity (which we'll call "veesquared", since it's going to keep coming up) is. Fortunately, height is (roughly, again we're oversimplifying air resistance). So, bottom line: there are squares involved at various stages, but the squares cancel out.


Does the drag really reduce the acceleration to a linear increase?

A linear increase in what? A constant acceleration is a linear increase in velocity in time - but yeah, drag ultimately reduces net acceleration to nothing (i.e. a constant velocity, namely what we call terminal velocity)


If so, then our velocity increase must be less than the pure 32 fs^-2 that I originally modelled... which means the 1dY per 10 ft drop rule isn't right, either: it's too high.*

That's pretty much how it goes. The problem is the same one with any rule - a trade-off between realism and bookkeeping. You don't really want to have to do exponential integrals in the middle of a game, so we need to build a table. I think lesser_minion was looking up the formulas needed for this

Anyway - some hard numbers - ignoring air resistance, and assuming gravity to be exactly 32 ft/s (it is actually slightly higher), your veesquared is going to be 2*32*10 = 640 ft/s per 10 feet. So 640 is the amount that translates to 1dY. I can't find the weight limits for size categories...

Ashtagon
2009-10-22, 10:52 AM
Really, falling for most games can be classed into three categories:

1) You fall into a pit/over a cliff, take token damage, and no meaningful way to go back. Classic rail-road move, and interesting done right.

2) You fall a small distance, take damage, and have to figure out how to get back up.

3) You fall from orbit. This is usually fatal.

The exact distance fallen is largely irrelevant for the first two cases.

The biggest issue I have with falling is not the hp damage, but the fact that you can walk away as if nothing had happened, provided it didn't actually kill you.

http://thepiazza.org.uk/bb/viewtopic.php?f=30&t=2545 is the thread I made elsewhere to compile my thoughts, although I'm still not entirely happy with the overall result.

lesser_minion
2009-10-22, 11:07 AM
Well, if you lose 99/100 hitpoints, then your DM has basically declared that the probability of you surviving that hit was 1%, and you got really lucky. The unrealistic part is actually the consistency with which you survive, but your character is a hero.

Falling from most orbits would actually prove fatal because of suffocation or vacuum exposure.

I'm pretty sure that falling actually becomes 'no chance of survival' for humans long before air resistance really makes much difference in any event.

Ashtagon
2009-10-22, 11:13 AM
Realistically, a person who is falling and not actively trying to maximise velocity will reach terminal velocity sometime around the 600-foot distance mark. Equally realistically, a fall of this distance should mean you die, no save allowed. Basically, air resistance shouldn't really be a factor in any environment that closely resembles the Earth.

lesser_minion
2009-10-22, 11:18 AM
Realistically, a person who is falling and not actively trying to maximise velocity will reach terminal velocity sometime around the 600-foot distance mark. Equally realistically, a fall of this distance should mean you die, no save allowed. Basically, air resistance shouldn't really be a factor in any environment that closely resembles the Earth.

You're forgetting the fact that a character above 5th level in D&D terms should be far more capable than any human being known to (have) exist(ed) in the real world.

A 20th-level character is a demigod - demigods don't die from such trivial things as falling off the Cliffs of Insanity.

Altair_the_Vexed
2009-10-22, 11:21 AM
The unrealistic "walk away from falls" trope that D&D and other HP systems have is why I added the CON damage.

My biggest difficulty was how to deal CON damage without being too harsh at low levels.
All of this number juggling with terminal velocities and energy and dice of HP damage aside - how could we model the difference between a fall that should kill you, and one where you might break something, and one where you're just going to be bruised and battered, but essentially okay?

Zeta Kai
2009-10-22, 11:43 AM
My biggest difficulty was how to deal CON damage without being too harsh at low levels.

I wouldn't worry about that. Just stick to a streamlined version of the RL physics. If it's too harsh for low-level characters, than DMs shouldn't risk flinging their low-level PCs off of cliffs, & low-level PCs should be avoiding cliffs in the first place (like I do). DMs who do so are sadists, & will find a way to kill their PCs, with or without realistic falling damage.

Random832
2009-10-22, 11:44 AM
The unrealistic "walk away from falls" trope that D&D and other HP systems have is why I added the CON damage.

Yeah, well, HP are unrealistic no matter whether it's falling or being stabbed with swords or being shot with arrows. The fix to that (if you want to fix it - characters above level 5 are basically superheroes) doesn't belong in a redone falling damage system. There are multiple redone HP systems out there - wounds/vitality, toughness saves, etc.


Realistically, a person who is falling and not actively trying to maximise velocity will reach terminal velocity sometime around the 600-foot distance mark. Equally realistically, a fall of this distance should mean you die, no save allowed. Basically, air resistance shouldn't really be a factor in any environment that closely resembles the Earth.

The real place air resistance would come in would be in determining how much time you have to do something about it. Falling from 600 feet should be _more_ dangerous than falling from 20,000 feet, if your character is even a little bit prepared.

lesser_minion
2009-10-22, 11:56 AM
The unrealistic "walk away from falls" trope that D&D and other HP systems have is why I added the CON damage.

My biggest difficulty was how to deal CON damage without being too harsh at low levels.
All of this number juggling with terminal velocities and energy and dice of HP damage aside - how could we model the difference between a fall that should kill you, and one where you might break something, and one where you're just going to be bruised and battered, but essentially okay?

First point: the current damage cap is far too low. While there are reported cases of people surviving landing at terminal velocity, the expected damage should be closer to 700hp (at the moment, it is 70)

I'd probably suggest using the fall distance to determine the DC of a check to land safety.

You could go with something like:

5 + number of spaces fallen (maximum 50) = DC to avoid taking double fall damage.

10 + 2 x number of spaces fallen (maximum 100) = DC to survive the fall with no damage.

Using either a Reflex save or a Tumble check.

Random832
2009-10-22, 12:02 PM
First point: the current damage cap is far too low. While there are reported cases of people surviving landing at terminal velocity, the expected damage should be closer to 700hp (at the moment, it is 70)

Yeah, well...

The solution:
Extraordinary Abilities (Ex)

Extraordinary abilities are nonmagical, though they may break the laws of physics.I would like to propose classifying having more than 5d8 hit points as an extraordinary ability for the purpose of this discussion, thereby putting "it should be impossible, on average, for a 20d12 barbarian to survive a fall from a great height" out of bounds. Otherwise everything has to be rebalanced, and you're basically inventing a new system.

If you want to rationalize it, say they're using ki, or hot blood, or spiral energy, or whatever you want to call it.

lesser_minion
2009-10-22, 12:06 PM
Yeah, well...

The solution:I would like to propose classifying having more than 5d8 hit points as an extraordinary ability for the purpose of this discussion, thereby putting "it should be impossible, on average, for a 20d12 barbarian to survive a fall from a great height" out of bounds. Otherwise everything has to be rebalanced, and you're basically inventing a new system.

If you want to rationalize it, say they're using ki, or hot blood, or spiral energy, or whatever you want to call it.

I'm perfectly happy with that rule, although it could cause problems with wizards polymorphing.

I actually pointed out already that a 20th level barbarian happens to be possessed of godlike endurance and toughness, and so shouldn't really die to something so trivial as a fall from the cliffs of insanity.

Altair_the_Vexed
2009-10-22, 12:11 PM
Okay - so let's simplify!

1dY per 5 feet fallen Cap the damage at 100 x die size for each size category* Grant a Tumble Check to take half damage DC 5 + number of dice (maximum 50). If the Tumble check exceeds 10 + 2 * number of dice, no damage is taken.

That's utterly different to what I came here with (except that different size categories take different damage, and terminal velocity - and thus maximum damage - is dependent on size).


*cause that's about the distance, coincidentally, when one reaches terminal velocity - according to my spreadsheet

Lappy9000
2009-10-22, 12:25 PM
{table]Size | Falling Damage Die | Maximum Falling Damage
Tiny|d3|9d3
Small|d4|16d4
Medium|d6|25d6
Large|d8|36d8
Huge|d10|49d10
Gargantuan|d12|64d12
Colossal|d20|81d20[/table]
I chose to avoid the table to show that the figures can be readily worked out from first principles.That table looks fine. Only get into the math if a player asks. I'd stick with "Your falling damage is based off size. I've got a chart! *display chart* That'd go over just fine :smallcool:

Altair_the_Vexed
2009-10-23, 06:23 AM
Edited the OP with the new revised version. (http://www.giantitp.com/forums/showpost.php?p=7170422&postcount=1)

Very different to my original plan, but I'd not had the excellent lessons in physics-to-D&D from lesser_minion then.

Random832
2009-10-23, 08:22 AM
Okay - so let's simplify!

1dY per 5 feet fallen Cap the damage at 100 x die size for each size category* Grant a Tumble Check to take half damage DC 5 + number of dice (maximum 50). If the Tumble check exceeds 10 + 2 * number of dice, no damage is taken.


Yeah, I just don't see the appeal of requiring it to do 100d6 - that only works if you accept the premise that a 20th-level character _shouldn't_ easily survive a fall from any height, and that premise is still under debate.

Also - even if terminal velocity is reached at 500 feet, that doesn't mean it is the same velocity that you'd hit in 500 feet without air resistance, which invalidates your reasoning for the 100.

I'll work on a table of my own and post it later

Also, you're wrong in another way - the cap _shouldn't_ be the same number of dice, because terminal velocity isn't going to be reached at the same distance in each size category.

----

EDIT: Table 1: Basic assumptions
{table]Size Category|Fine|Diminutive|Tiny|Small|Medium|Large|H uge|Gargantuan|Colossal
Typical Mass|0.04|0.29|2.34|18.75|150|1200|9600|76800|6144 00
Typical XSA|0.0390625|0.15625|0.625|2.5|10|40|160|640|2560
Calculated v∞|28|40|57|80|113|160|226|320|453
Calculated v∞|800|1600|3200|6400|12800|25600|51200|102400|20 4800
Dice Cap (v∞/640)|1|3|5|10|20|40|80|160|320
Die Size|1|2|3|4|6|8|10|12|2d8[/table]

150 and 10 were arbitrarily chosen, partially to match medium creatures to the existing cap of 20. The other size categories were worked out by successively multiplying or dividing by 4 and 8.
The die sizes were arbitrarily chosen, to reflect the increase in mass but also the fact that they can likely absorb more energy in a way not fully reflected by their hit point total.
640 was chosen to match the existing rule of 1dY for every 10 feet fallen for low heights.
v∞ is calculated as = √(2mg/ρACd), where g is the acceleration of gravity (32) ρ is the density of air (.075 lb/ft) A is the cross-sectional area, and Cd is the drag coefficient (assumed 1)

A table matching heights to v for each size category is forthcoming (as soon as I can work out a formula)

boomwolf
2009-10-23, 09:01 AM
I believe you are forgetting something...

Why on earth would a colossal creature take ANY damage from a 20 ft. fall? not to mention more then medium creatures?

Its like falling 2 feet for a human, its insignificant.

He probably takes his own feet higher then that for a mere step.

And fine creature takes no damage? why so? the 20 ft. for him is quite a distance.

Sure, impact energy grow with size, but bigger creatures spread it across bigger surface and they have bodies designed to be using much more energy for movement.

By the logic you are applying, a good tactic to kill a colossal creature is to make him walk around. the energy from his mere steps will end up killing it quickly.

I'd stay with the linear system. simpler and makes more sense sometimes.

Random832
2009-10-23, 09:13 AM
I believe you are forgetting something...

Why on earth would a colossal creature take ANY damage from a 20 ft. fall? not to mention more then medium creatures?

Its like falling 2 feet for a human, its insignificant.

He probably takes his own feet higher then that for a mere step.

And fine creature takes no damage? why so? the 20 ft. for him is quite a distance.


They're going to have a lot less energy from weighing less, and they're not going to be going any faster (and may be going significantly slower) than a human who falls 20 feet. It being "quite a distance" means they're going to be disoriented if anything, not that they're going to take damage.

Your idea of scaling based on how much distance you intuitively think is "significant" compared to the size of the creature is wrong - gravity pulls with the same acceleration whether you're 6 feet high or 60 or 6/8 of an inch. (and even if it didn't, it certainly wouldn't be MORE for smaller creatures and LESS for larger ones)

And anyway, the dice sizes were meant to take all that into account. If they were simply scaling to the raw energy, it wouldn't be averages of 3.5 4.5 5.5 6.5 9, it'd be 3.5 28 224 1792 14336.

But maybe it should be a flat d6. I'm still not 100% certain on die sizes.

And the falling damage wouldn't even kick in for walking around since you're never completely off the ground.

Ashtagon
2009-10-23, 09:21 AM
Conceptually, a fall from a height that ends up accelerating the person to a speed of X mph is equivalent to being hit by a bus moving at that same speed.

Is there any way to integrate falling damage with collision damage?

Random832
2009-10-23, 09:30 AM
Conceptually, a fall from a height that ends up accelerating the person to a speed of X mph is equivalent to being hit by a bus moving at that same speed.

Is there any way to integrate falling damage with collision damage?

I'm not sure how often anything like being hit by a bus is going to come up (you've got a flat unyielding impact surface that is quite unlike being run over by horses), but just for fun - given the RAW premise of 1d6 per 640 v...

{table]dice|min.v|max.v|min.ft/s|max.ft/s|min.mph|max.mph|min.kph|max.kph
0|0|320|0|18|0|12|0|20
1|320|960|18|31|12|21|20|34
2|960|1600|31|40|21|27|34|44
3|1600|2240|40|47|27|32|44|52
4|2240|2880|47|54|32|37|52|59
5|2880|3520|54|59|37|40|59|65
6|3520|4160|59|64|40|44|65|71
7|4160|4800|64|69|44|47|71|76
8|4800|5440|69|74|47|50|76|81
9|5440|6080|74|78|50|53|81|86
10|6080|6720|78|82|53|56|86|90
11|6720|7360|82|86|56|58|90|94
12|7360|8000|86|89|58|61|94|98
13|8000|8640|89|93|61|63|98|102
14|8640|9280|93|96|63|66|102|106
15|9280|9920|96|100|66|68|106|109
16|9920|10560|100|103|68|70|109|113[/table]

Ashtagon
2009-10-23, 11:18 AM
I do my maths in metric, then convert back, so...

G = 9.0665 m/s2 (~32.174 ft/s2)
T = time spent falling (assumption: acceleration is constant until terminal velocity is reached).
Vf = final velocity
Va = average velocity
D = distance fallen

H = Standard human mass = 68 kg (~150 lb) (and yes, this is the real-world average, based on official USA statistics).

final velocity = T x G
average velocity = T x G x 0.5
distance fallen = T x Va
distance fallen = T x T x G x 0.5

With distance (plus constants) as the only known quantity...

falling time = (D x 2 / G) ^ 0.5
average velocity = (D x 2 / G) ^ 0.5 x G x 0.5
final velocity = (D x 2 / G) ^ 0.5 x G

impact energy = H x Vf x Vf

I'll leave it to the bored reader to calculate numbers, although I did run a spreadsheet.

tl;dr version: Impact energy increases linearly with distance fallen, and is approximately 4065 joules per 10 feet fallen (why yes, I am mixing units here).

DracoDei
2009-10-23, 11:44 AM
As both a gamer and someone with a Bachelor's Degree in Mechanical Engineering I approve of the O.P. as it currently stands, and/or Random832's suggestion.

Ashtagon
2009-10-23, 11:47 AM
DracoDei (and anyone else competent), can you confirm or disprove my maths?

Edit: One issue with using energy as the basis of damage is that most rules systems seem to take the square root of the energy as the base value on which to calculate damage. This is true of Traveller's Fire, Fusion & Steel weapon design systems, and equally true for GURPS Vehicles' weaponry chapter.

Edit2: (impact velocity in mph)^2 x 29.914 = equivalent falling distance in feet, just in case you ever need to know what it feels like to be hit by a bus.

Edit3: Basing damage off the root of the impact energy is the same as basing it linearly off the force (in newtons) of the impact.

Random832
2009-10-23, 12:52 PM
Edit: One issue with using energy as the basis of damage is that most rules systems seem to take the square root of the energy as the base value on which to calculate damage.

So what you are saying is, one issue with doing things one way is that other systems do things a different way?


Edit3: Basing damage off the root of the impact energy is the same as basing it linearly off the force (in newtons) of the impact.

energy = mass length2 time-2
force = mass length time-2
square root of energy = mass0.5 length time-1

force of impact depends on how hard the surface you're falling into is, which does not affect impact energy or its square root - though it would arguably affect damage.

This weekend I'll see if i can piece together a formula to take air resistance properly into account while still having the table in terms of height vs energy.