*The Dragon*#88 (August 1984) had an infamous set of articles on the physics of falling damage. Some fans of the game maintain that this investigation of math and physics is unnecessary to a fantasy game. However, I’m on the other side of this argument; if there must be some base rule for a mundane activity like falling, then I see no reason not to “dial it in” correctly. Surely there’s no disadvantage to the parties who claim to simply not care one way or the other. Hence, I consider these articles to possibly be the very zenith of D&D system scholarship as it could be practiced.

### Dragon #88: Parker vs. Winter

The issue in question appeared in the context of the OD&D and AD&D book rule that falls do a linear 1d6 per 10 feet; and a recent article and update in UA in which Gygax claimed this was a typo, and should have been a far more brutal 1d6 cumulative per 10 feet (e.g., 3d6 at 20’ and so forth). One of the Dragon #88 articles is by A.A. Parker on “Physics and Falling Damage”, arguing that damage should be proportional to velocity, and thus more violent at the start, but scaling down somewhat with greater height (e.g., starts at 3d6+1-5 for a 10’ fall; reaches 20d6 with a 260’ fall). The other article in that issue was by Steve Winter called “Kinetic energy is they key”, whose conclusion is that kinetic energy, and thus damage, is linear with height – thus arriving back at the 1d6/10’ rule we started with.Let’s look a little more closely at the argument with the advantage of time. Interestingly – something I tend to forget – is that Parker was apparently conscious of Winter’s following rebuttal, and spends several paragraphs and a sketch (above) addressing the kinetic energy argument, and ultimately rejecting it (p. 14-15). But his culminating argument is fairly weak:

At best, all of Parker's counters here are equally applicable to his own velocity-based thesis. But worse: there is in fact a physical law which effectively asserts what he claims is missing, and it’s called the “work-energy principle”: per Wikipedia, “No physical law exists that says kinetic energy is the direct cause of physical injury. We know that there is some relationship between the two – because the more kinetic energy a person transfers to the earth, the greater his injuries are. But no law states that this relationship is linear, or that all the factors involved in kinetic energy relate to the injury. It may be, then, that some part of kinetic energy relates linearly to falling damage. Since no formula exists to tell us what part this might be, we have to use our intuition to determine the crucial property.

*the work done by all forces acting on a particle (the work of the resultant force) equals the change in the kinetic energy of the particle*”. Assuming a body goes from some initial velocity to a full stop, then the work on the body is exactly equal to the kinetic energy, given by KE = ½ mv^2. (where

*m*is the mass of the body, and

*v*the velocity). So we see that work done is not proportional to velocity as Parker argued; it is proportional to the square of the velocity, and hence directly proportional to energy, as Winter asserted.

### Collision Model Data

Consider this from a different perspective: It’s now easy to access data sources such as those used to model pedestrian injuries and fatalities from auto accidents (a fairly good analog to a body hitting solid ground at high speed). See Richards, “Relationship between Speed and Risk of Fatal Injury: Pedestrians and Car Occupants”, Department for Transport, London (2010). Data for pedestrian fatalities, mapped against auto speed, follow a characteristic sigmoid curve. Several proposals for model formulas for probability of death are given; the simplest, by Rosen and Sander is: P = 1/(1 + e^(6.9 – 0.090v)), where*v*is the automobile’s velocity, and

*P*is the probability of a fatality (p. 14). Below we attempt a linear regression against speed itself (as a test of Parker’s theory), and also speed^2 (proportional to kinetic energy, as a test of Winter’s theory).

Taking probability of death as proportional to damage to the body, then it’s rather obvious visually that the better fit belongs to the speed^2, i.e., kinetic energy measurement. The coefficient of discrimination to this fit is R^2 = 0.977, that is, kinetic energy serves to explain 97.7% of the variation in mortality asserted by the model.

Interestingly, the auto-collision industry uses a mathematical model for estimating speed from observed damage, called CRASH (Computer Reconstruction of Automobile Speeds on the Highway) which explicitly takes as a first assumption that depth of crush is a linear function of velocity (or momentum). However, at least one presenter admits that this is an approximate model only: see McHenry, “The Algorithms of CRASH”, McHenry Software, 2001 (p. 19-20):

Combined assumptions of (a) linearity of ΔV_c (i.e., the ΔV preceding restitution) as a function of residual crush and (b) a ΔV_c intercept near 5 MPH, at which no residual damage occurs, have served as the basis for extrapolations outside the range of available test data... The use of linear relationships may be viewed as a simple empirical process for interpolation and extrapolation of the results of staged collisions... it appears that bilinear fits might yield more accurate application results when a large ΔV range is included in the fitted data.

### Terminal Velocity

Many editions of D&D (starting with the AD&D PHB, p. 105) have a rule capping fall damage at 20d6, and often times this is taken as a simulation of terminal velocity of a falling man (as in the Parker & Winter articles). One problem arises, however: in the core rule this occurs after a 200’ fall, when in reality terminal velocity of a falling person isn’t reached until much later. Compare also to Carl Sargent's revised falling rules in PC2*Top Ballista*(1989, p. 61), in which he is clearly looking at real-world speed/distance of falls, and greatly reduces damage to have the 20d6 max occur at real terminal velocity.

Terminal velocity for a falling man in a stable, belly-down position is around 120 mph, or about 180 feet/sec (Wikipedia). Due to the asymptotic nature of gravity vs. wind resistance, a person only reaches 50% of this speed after 3 seconds or so (about 140 feet), 90% of terminal at 8 seconds (800 feet), 99% of terminal at 15 seconds (over 1500 feet), and so forth (in a theoretical sense, one never actually reaches the terminal speed; it's only a limit). See the excellent chart by Green Harbor Publications, 2010; to my eye, the “inflection point” in the graph is at around a 6-second, 500-foot fall, at which velocity is about 80% of terminal. In core D&D terms, this would argue for max falling damage of around 50d6 or something like that.

Let's compute for more specificity on that point. For simplicity, I define a new energy unit, the

*footman force*, as one man × ft^2/sec^2 (compare to the foot-pound force). As a preliminary, we compute the speed from a 10' fall, assuming that for such a short height, air resistance is negligible: sqrt(10)/4 × 32 = 25 ft/sec. This allows us to compute the kinetic energy from the 10' fall: KE = 1/2 mv^2 = 1/2 (1 man)(25 ft/sec)^2 = 312 footman forces. On the other end of the continuum, where air resistance is total, we are told that terminal falling velocity is about 180 ft/sec; and in this case the kinetic energy is KE = 1/2 (1 man)(180 ft/sec)^2 = 16,200 footman forces. The ratio between these two energy amounts is 16,200/312 = 51.9; that is, very nearly 50 times the energy (and we would argue, damage) between a 10' fall and one at terminal velocity.

Again, the calculations above are shortened a bit by assuming the mass (

*m*) of the falling body is simply "one man". It could be an important point to observe that smaller creatures will generate less energy (damage) from a fall, and larger creatures more. Wikipedia quotes the biologist J.B.S. Haldane as writing:

That said, we ignore this detail at the present time and consider a rule only for approximately man-sized creatures.To the mouse and any smaller animal [gravity] presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom, it gets a slight shock and walks away. A rat is killed, a man is broken, a horse splashes. For the resistance presented to movement by the air is proportional to the surface of the moving object.

### Falling Mortality

The real-world statistics of falling mortality are expressed in terms of “median lethal distance” (LD50), that is, the distance at which a fall will kill 50% of victims (who are presumably normal adults). Smith, Trauma Anesthesia, p. 3, asserts that LD50 is around 50 feet (4 stories). Wikipedia asserts that LD50 for children is at a similar height; 40-50 feet (one might think that children are less durable than adults, but note the observation on size/mass and damage above). Dickinson, et. al., in “Falls From Height: Injury and Mortality” (Journal of the Royal Army Medical Corps, 2013) notes that LD50 varies greatly by injury type: about 10.5m (34 feet) for those who land on their head or chest; about 22.4m (73 feet) for those who do not. (To me, this fact argues that falls should get some kind of binary saving throw, possibly save-vs-stone for half damage? And recall that the first falling rule in OD&D Vol-3 indeed gave a saving throw using a different mechanic.)So: Around 50 feet kills about 50% of human victims. Note that this is actually much more generous than the standard D&D rule, which in its simplest form will kill about half of normal men with just a 10’ fall (1d6 damage vs. 1d6 hit points). It broadly contradicts Gygax’s advice in OD&D Vol-3, p. 6 (“there is no question that a player's character could easily be killed by falling into a pit thirty feet deep”). To say nothing of Gygax’s intuition at the time of UA that such damage should be massively upgraded with a cumulative rule.

### Considered Rule Edits

Here we consider a few rule edits based on the mortality data seen above. Among the rules edits considered are:- A save-vs-death at zero hit points; this has been the existing OED house rule for some time (recently edited so that any "overkill" damage becomes a penalty to the save). If you like, consider this to be broadly analogous to permitting negative hit points before death.
- A save-vs-stone for half damage; this is inspired by the medical findings that there is a binary difference in mortality depending on whether a victim hits their head/chest or not (specifically, almost exactly a 50% change in LD50 mortality).
- Possibly reducing the damage dice to something like 1d6 per 20 feet.

On the one hand, the basic D&D rule is unrealistically harsh: 1st level characters (and normal men) have 100% mortality from a 50' fall. Even if we engage any one of our cushioning rules, then it is 3rd-level characters who have around a 50% mortality rate (as shown in the first chart).

But on the other hand, if we do calibrate the system so that 1st-level characters have a 50% mortality rate (as in the real world), then that requires engaging all of our proposed edits -- reduced base damage, a save for half,

*and*a save vs. death at zero hit points (as shown in the second chart). And then as a result, any higher-level characters have effectively negligible chances of perishing from a fall of that height (further down the same column).

Let's take a case study from the pulp literature: in the short story "The Scarlet Citadel", Howard relates a scene in which Conan grapples the sinister and powerful Prince Arpello and hurls him from a 150' high tower, upon which, "the body came hurtling down, to smash on the marble pave, spattering blood and brains, and lie crushed in its splintered armor, like a mangled beetle" (Ch. 4). If we assume this worthy is a 9th-level Lord (he's described as a capable fighter, a veteran of many campaigns), then we can re-run the simulation at this greater height. If we use the latter fully-cushioned rule proposal, then mortality from this fall would only be 2% -- clearly unacceptable. If we keep the 1d6/10' rule but allow the two saves, chance of death would be 35%; with just the half-damage save 43% (I think both of those are still too soft); with just the existing OED save-vs-death mortality is 72%, which feels about right (no saves at all would give 85% mortality).

### Conclusions

In summary: Calibrating the rule so that normal men have a real-world LD50 at 50' requires a rule so generous that it completely violates our intuition and literary examples for higher-level characters falling from truly stupendous heights. The best we can do is massage something acceptable in the middle. And it turns out that the classic 1d6/10' damage, with a single save of some sort (like: OED save-vs-death, or some negative hit point allowance) does give a vaguely reasonable result around 3rd level or so. Considering that this effectively requires no modification to our existing ruleset, it seems like living with this is among the best options.It also seems like a recurrent theme that many parts of the D&D system are most "realistically calibrated" for characters of around 3rd level; perhaps this gives extra support for starting campaigns at around that level (as Gygax did in later years; see ENWorld Q&A thread, date 11/19/04). See also: Environmental Rule of Three. I'm comfortable with existing OED save-vs-death at zero hit point representing the variation in whether the victim lands on their head/chest or not. And I think I'm now compelled to bump up the maximum fall damage to 50d6 to reflect true terminal velocity (which neatly takes care of the need for extra saves-vs-massive damage or the like, since everyone up to a lesser god gets equally pancaked from arbitrarily high falls).

Open question: In pulp heroic literature what other examples of survivable falls do we see for figures like Conan, Fafhrd, Elric, etc.? Perhaps we could use that to “dial in” the rule better, if necessary.

What if the first 50' deals 1d6 and every 10' after that adds another 1d6?

ReplyDeleteI've spent many an hour going down (falling down?) this particular rabbit-hole myself.

ReplyDeleteI think the fundamental problem in reconciling reality with D&D-like games is that health in D&D is in some respects binary -- you are either dead or effectively so (0 hp or less), or else you are "fine" (1 or more hp, which might be "near-death" in some sense but which doesn't impair your ability to act in any way. While LD50 may be 50' for a typical human, I feel confident in asserting without evidence that the vast majority of those that are NOT killed by a 50' fall are nevertheless incapacitated (broken/shattered legs, for example).

There are also cases where people are killed by very short falls (a few feet), or survive falls at terminal velocity.

If you want high-level figures to still be as vulnerable to death from falls as commoners are (or nearly so) then I think the better approach would be to always just have a save vs. death for any fall, with the difficulty/DC of the save depending on distance. Save bonuses go up much more slowly than hit points, so "name-level" figures would still be intimidated by say a 50' or 100' drop.

I agree that many non-fatal falls should result in incapacitation, and that's taken for granted in the post above. Many of those surviving will still be at unconscious 0 hit points (either by my save mechanic, or negative-hp rule, as one prefers) and so incapacitated. If anyone wasn't yet playing with one of those cushioning rules, I would highly recommend they make the switch (so as to account for exactly this either-fine-or-dead issue in book OD&D).

DeleteI've never read the series, but from my understanding of it from other sources, Mervyn Peake's "Gormenghast" series featured several defenestrations and hurlings from castle spires. Mostly fatal, from what I've gathered.

ReplyDeleteThe only other literary example that springs to mind is Holmes & Moriarty at the Reichenbach Falls. Dunno if that's helpful 'cos only one of the participants of that scuffle actually met his end at the bottom of the falls, and I don't think Doyle was very explicit about it.

Cool references! I did find two others by Conan: (1) dropped maybe 2 floors through a trap door, lands on his feet, runs off to further combat (Slithering Shadow, Ch. 3), (2) dropped through a trapped door into a rushing black river, swims off with sword in teeth basically fine (Jewels of Gwahlur, Ch. 1).

DeleteFrom Slithering Shadow: "He did not fall any great distance, though it was far enough to have snapped the leg bones of a man not built of steel springs and whalebone. He hit catlike on his feet and one hand, instinctively retaining his grasp on his saber hilt..."

DeleteConan is essentially a panther in sandals and a chain mail hauberk. Of course he lands "catlike on his feet" and is made of "steel springs and whalebone". He's fantasy Chuck Norris.

DeleteHmm, can't think of any examples in pulp fiction to go by.

ReplyDeleteI recall a similar discussion around Mutants and Masterminds about how deadly a fall should be in a system that lets you shrug off a blast of lava to the face.

This is one of those issues that also falls back on to "What is a Hit Point?" and "Are high level characters still normal/mortal?" I would almost be inclined to say that Falling, like swimming in lava, is one of those things that should not interact with HP.

One could argue that HP are just for combat, turning away "hits" and whatnot. Or that after 6th, 10th, whatever level, you are now "mythic" and not always subject to the same laws that govern mortal folk (see our comments around Beowulf and swimming/breathing)

Otherwise I am inclined to say, in the absence of a greatly improved rule, stick with the original.

I used to think that myself (that environmental stuff should not be linear with HP), but I've come around in the other direction over the years. On the hand, the pulp literature seems to give characters extra-survivability in those cases (see my Conan fall above); on the other real-world hand, we have the many examples of spectacular survivorship by certain lucky or iron-willed people (far more than twice the average in falls, lack of air, survival with no food, e.g.). So I've grown more comfortable with high-level PCs being extra-tough in all kinds of pursuits.

DeleteI confess the sin of not reading the linked references before commenting, but: do the sources on falling-damage LD50 also consider the surface landed upon? Perhaps many of the survivors landed in water, while many of those who died landed on concrete?

ReplyDeleteI haven't seen it broken down that way. Since the majority of falls are construction accidents (off a ladder or roof of a home), I think that cases of falling into water is probably negligible. Also, cases tend to only get documented where there is some injury.

DeleteLanding in water from great height is better than landing on concrete, but not all that much better. Water resists impact surprisingly well.

DeleteAbsolutely true. On the other hand, the pulp sensibility/examples are that it's a lot safer, so if one made it safer in their game, that's a defensible design choice.

DeletePatrick Stuart's excellent take on the Underdark, "Veins of the Earth", has the following rules for falling damage:

ReplyDeleteFEET FALLEN DAMAGE

10 1d6

20 1d6 x 1d4

30 1d6 x 1d6

40 1d6 x 1d8

50 1d6 x 1d10

60 1d6 x 1d12

70 1d6 x 1d20

80 1d6 x 1d50

90 1d6 x 1d100

He has this to say about it:

"This is longer and more annoying than the standard falling damage chart, and harder to use. But it is a more consequential, and less predictable system which makes falling both more dangerous for high level characters and also possibly-survivable for low level ones."

What do you think about this?

This is quite relevant to me, as I'm currently running Patrick Stuart's Deep Carbon Observatory, and the group are just about to head down into the Veins. In the last session, they dropped the white Giant off the bridge into the darkness below, and it's important to know if it survived. On the one hand, it's a huge creature, liable to take more damage, on the other, it is entirely cartilaginous, and fell into water. And it had a lot of HP and a very good ST.

DeleteFrom 10-60' it seems like the same average results with greater complication. (To me, the Nd6 is variable enough.) The 70-90' extra damage is like Gygax's cumulative rule: exponentially more aggressive than reality. As in the post above, if 10' is 1d6, then the 50d6 average shouldn't be reached until 500'.

DeleteOf course, if one thinks that HP simply shouldn't provide linear resistance to environmental factors like falling (e.g., Baquies above), then that's a different story -- but not what Stuart seems to be arguing.

Oh bypassing HP wasn't my personal viewpoint, just exploring a line of thought. It is probably valid for some settings, game types.

DeleteI am comfortable with high level folk surviving/doing crazy stuff.

Thanks for the clarification!

DeleteWhat would be the effect of the character's encumbrance upon the kinetic energy of the impact? Would it increase lethality?

ReplyDeleteWhat if the falling damage were something like "1d6 + level per 20 ft."? This might balance the "HP are just for combat" and "extra survivability" arguments.

...

My hastily made spread-sheet shows that the average-HP and average-falling-damage intersect at 5th level (with the above formula)(for a d6 HD). Not knowing the math behind the saving-throw calculations, I have to leave the rest as an exercise for Dan.

Well, since kinetic energy is proportional to mass (KE = 1/2 mv^2), having more encumbrance does increase KE proportionally. Although then one debates how much that extra stuff soaks up the KE at the end of the fall (e.g., sack of rations and potions squashing and partly breaking your fall). Maybe an overall wash in terms of damage to the person?

DeleteEncumberance was my thought as well, that perhaps Gary Gygax had in mind the weight of the armour that typical adventurer is wearing when falling and so thirty feet may have 'felt' about right. Certainly a knight in plate would have a different LD than a man in a tracksuit.

ReplyDeleteinterstingly, right after putting some numbers down for falling rules, I get my computer and ... here is this post!

ReplyDeletemy conclusion, based of simplistic calculation is that the rule of 1d6 per 01 feet is the average of linear and squared damage for speed (so I decided to postpone any more calculations).

If you want to poke more holes in my logic, here it is:

-I simplified the situation by deciding that the acceleration would be "instant" and "after the fall: for example, after 1 seconde, you're going exactly 30 feet/second. I used this as base damage (3d6).

If you go twice as long (2 seconds), you now move 60 feet/second, for a total of 90 feet (9d6 per AD&D, 6d6 if damage proportional to speed and 12d6 if you square the speed (twice as fast, so 4 times 3d6)...

anyways, as I mentionned: extremely simplistic, as there is no wind resistance. I decided for 3d6 as the base because it's the amount of damage that would seriously hurt a first level character (I give some bonus HP at first, from Tao of D&D's rules). Without being able to decide, I averaged the results, and we're back to 1d6 per 10 feet.

Maybe they were unto something: this rule is very gameable, players can quickly judge their survival odds and the results are good enough.

and by the way, I love your complex mathematical analysis of rules, good food for thoughts

Cool! I agree with that latter point: game-able and tractable for players to estimate and make a decision about in-game. It is funny that more complicated analyses bring us back to a simple 1d6/10' in many cases.

DeleteI can also see the attraction of some more hit points at 1st level; there are times I wish that the 1st-level increment was 3d6, so that progression to 2nd level wouldn't be so radical. But then we'd have to possibly re-gauge all the combat damage, because a legitimate criticism would then be "a sword thrust should have at least a chance of killing a normal man".

I wonder if a good way to handle this might be to shift the focus onto the ST and away from HP.

ReplyDeleteWe know the LD50 is 50 ft, so set the average Death save (or whichever save you feel is most appropriate) for level 1 characters to give a 50% chance. Say, for the sake of argument, that a level 1 character saves on a 15+ (so a 6 with Target20, or 15 with straight OD&D). Then a 50 ft fall needs a +4 to the save.

Sticking with Target 20, there are then 4 outcomes:

1) Fail (0-9): lose 2x max HP (probably resulting in strawberry jam on the ground).

2) Fail (10-19): lose 1x max HP (maybe killing you outright, but perhaps surviving with negative HP, albeit unconscious and dying).

3) Pass (20-29): lose 1/2x max HP (you might walk away unscathed, providing you haven't suffered too much damage already)

4) Pass (30+): no damage.

However, the save needs to be modified to account for height fallen. Maybe -1 per 10 ft, so a high level character can fall about 50 ft further than a low level character. This could also be nonlinear up to however far it takes to reach roughly terminal velocity.

Finally, modify the save depending on size. Small creatures take less damage than big creatures, so +0 for human sized characters, then +2 for each halving of size, or -2 for each doubling of size (or possibly a +2, +5, +10, +15 progression).

As a quick test, a level 1 mouse falls 100 ft, that's 16 = 6 - 5 (height) + 15 (size). Ends up at zero HP 20% of the time, half HP 50% of the time, and full HP 30% of the time. A level 1 elephant falls 10 ft, that's 0 = 6 + 4 (height) - 10 (size). The elephant splats 50% of the time, and is knocked unconscious the other 50%. A bit of tweaking probably necessary, but it seems reasonable given a cursory look, and no worse than basing things on HP.

Ugh, I forgot to add in the +4 I mentioned at the start!

DeleteIn the falling damage discussion I always have wondered, why should damage scale linearly with height?

ReplyDeleteYes, work, and kinetic energy, I see that, but why is damage linear with energy? I would have guessed that damage is linear with the (resultant) force.

It is force that breaks bones and splits skulls, right!?

So, I sat down and started to scribble:

Force (F) determines how fast (time t) momentum (p) can be changed: F = dp/dt. Momentum is mass times velocity (p = mv).

So, from this I would have guessed that damage scales linearly with force which scales linearly with momentum (and thus velocity for a given mass)!?

But, then it finally dawned upon me: You also have to take into account how much there is available for the force to act on the body.

Hmm ... how much is available? Well, the time between hitting the ground first with a body part until the full stop where the body lies (scattered?) on the floor. During this time the center of mass covers a certain height and this height is independent of speed, but the amount of time to cover this height does scale with t~1/speed.

And thus:

F = dp/dt which scales as v / (1/v) = v^2.

So, the force scales with v^2 and thus linearly with KE (and thus height), after all!?

And, yes, now I also understand how the work argument comes into effect: work equals force times distance (W = Fx). To drop the velocity to zero, you need a force whose work nullifies the total KE. Since x is constant, as explained above, the force must scale with KE (and thus height).

In short, I think the fundamental quantity is force and not KE per se. However, the force does scale with KE, so there is that.

I guess I am not smart enough to see the connection right away, but with the above reasoning, I agree with your statement!

Thanks for posting that. Feels nicely Socratic. :-)

Delete:)

ReplyDeleteJust after posting, I realized the following short-cut:

KE to overcome = Work = Force * distance.

The distance is the the height you cover between firstly touching the ground and the final stop. This distance is independent of the falling height, and thus force must scale with KE.

Forgive me if you had already alluded to this in your post and I just missed it, but would it not be sensible to multiply the damage by the hit dice of the creature? Then you could have a flat die progression and calibrate it to what the hp of a normal man is in your system. In mine, a healthy adult has 4 or 5 hit points. Taking into account the save vs death, I'd assign 50' a damage of 1d10, as 5.5*0.65+2.75*0.35=4.5375.

ReplyDeleteSo, for my hp values, I'd have a progression something like

10': 1d2

20': 1d4

30': 1d6

40': 1d8

50': 1d10

60': 1d12

70': 2d6

80': 2d8

90': 1d20

100': 2d10

The short story for me is that in the past I used to work such systems. More recently I want to honor the higher hit dice as really super-humanly more durable to falling, exposure, cold, etc., and so more simply not adjust damage in that way.

Delete