Falling Revisted

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:

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.

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, “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:

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.

That said, we ignore this detail at the present time and consider a rule only for approximately man-sized creatures.

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.

If we consider such changes, we need to keep an eye on the overall effect at both low and high levels. A simulation in Java code was run for the event of falling 50', for each of the proposed edits, for characters of varying levels (trials N = 10,000 for each case). Results are shown below (highlights at around 50% mortality):

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.

Addendum 2021: Regarding the 1984 Parker (velocity) vs. Winter (energy) debate over what is a linear function of damage, see the exact same question asked in 2014 at Stack Exchange: Physics -- without a clear resolution or selected answer at this time. (!)