Falling Re-Revisited

For some people, the dueling physics articles about falling damage in Dragon Magazine #88 (August 1984) were the worst thing ever, but to me they're the best. I did a deep dive into them on a number of points two years back.

Somehow I got sucked into them again, almost unbelievably, as I was preparing for a Marvel Super Heroes FASERIP game that we ran on the Wandering DMs Solo Play show the last few weeks. (The MSH Charging rule was massively changed between Basic & Advanced versions; which is best? Well, the best physical model I could think to base it on was falling, and as a result I sort of house ruled all of time-distance-height-falling scale rules as I'm wont to do.)

Previously I've was always sold on Steve Winters short argument "Kinetic energy is the key", in that kinetic energy is linear with height, and so falling damage would be as well: say 1d6 per 10 feet. Simple and nifty.

But for some reason it finally dawned on me that both articles in Dragon #88 entirely ignore an important issue: air resistance. Maybe they didn't have sufficient computing power to model it at the time?

Now, it's close to common knowledge that energy and velocity are related by the formula E = ½mv² e.g., this is brought up in almost any discussion of the danger of high-speed car collisions. The fact that Parker's main article actually quotes this formula and then explicitly discards it (in an attempted counter to Winter) is so incredibly wrong that in retrospect that I almost feel physical pain at the embarrassment from it.

The problem with using kinetic energy to determine damage is this: kinetic energy is a function of the square of velocity. Everyday physics (the classical mechanics) is very much intuitive. It does not make sense that the square of velocity linearly relates to falling damage; it does make sense that velocity itself directly relates to damage. When a person hits the ground at speed 2x, he should take 2d of damage -- not 4d. Therefore, we should feel free to discard the concept that kinetic energy is linearly related to falling damage.

Ouch. It may "not make sense" to Parker, but it's the fact anyway. On the other hand, by relying solely on potential kinetic energy at the start of a fall, Winter makes something of the reverse error -- ignoring the fact that a lot of the kinetic energy will be scrubbed off by (non-damaging) air resistance. In that regard, we really should look at the velocity at the end of the fall, and convert (by squaring!) to energy actually released at that point. Additionally: It wouldn't make sense for damage/energy to really be increasing regularly with every unit of distance, and then have some abrupt point where that suddenly stops because terminal velocity was reached (as Winter posits).

I'm really astounded that this exclusion of air resistance in both articles never occurred to me before, and I'm scratching my head at how I overlooked it for so long. Let's see what we get if include that, and whether or not we want to observe that in a game system.

Somewhat surprisingly, Casio.com has a nice online calculator for falling with wind resistance that lets you compute time and velocity for a single fall at a time; the comments show it's mostly used by people in their TTRPG games. The formulas used look like the following (I'm not sure where the derivations come from, but the results closely match the figures quoted from Sellick's Skydiving in Dragon #88.)

Here's a tabulated set of results from those calculations:

If we take the 10' height as the basis for damage assessment, then the "Energy Multiplier" column is effectively the number of dice we should roll for damage from the indicated fall. Some observations: Due to the increasing effect of air resistance, the increase in velocity, and hence damage-per-unit tapers off with greater heights. For example, at 10' we roll 1d6, 20' 2d6, 30' 3d6, etc., as we're used to. But at 80' it's only 7d6; at 200' it's only 16d6. At extreme heights (probably pretty unlikely in most games), the damage-per-unit-height becomes effectively zero.

Here, we gradually approach terminal velocity as a limiting value, which makes a lot more sense than Parker or Winter with an abrupt cutoff at around 200' height. It's kind of interesting to see that even a 10' tall drop gets you to 15% of terminal velocity (last column), while 200' is only about 60%; 500' is about 80%, and not until a 2,000' drop is reached do you get to 99% of terminal velocity. But that smoothness is certainly more what I'd personally expect from the real world.

The damage increment at terminal velocity is 47 times the 10' fall; say about 50d6, which is what I've previously written into my OED house rules. Now, should we actually implement this slow drop-off in damage in-game? Since it's nonlinear, this is a case where the only legitimate way to do that is with a table (or a digital app, ugh), for which I don't think I'd want to spend time or space. So likely I'll just keep with the linear approximation -- it's pretty spot-on for distances up to 100' or so. Whereas the 500' fall should really only give about 30d6 damage, for me that's where I'll top out at the terminal 50d6 (being aware at the difference from reality).

Another observation is that when accounting for air resistance, weight matters. That is: as per the classic experiment by Galileo, if neglecting air resistance, we're used to saying that everything drops at the same acceleration, regardless of weight. But the formulas above actually do include a factor for weight (mass, m), and you get different results at different values. This reflects the observation previously by J.B.S. Haldane that for long falls, "A rat is killed, a man is broken, a horse splashes". For what it's worth, in my table above, I presumed a man of 150 pounds (68 kg, or 11 stone).

An additional item for me is that I recently broke down and decided to give a save for half damage on any fall (which both models the real survival rates for normal people a bit better, plus the bimodal rate of  whether someone hits their head or not). The downside is that for me this might trigger as many as 3 saving throws: (1) to avoid the fall, (2) for half-damage, (3) for a death save if hit points are depleted. But I think I'm willing to live with that in light of the other simulationist advantages.

Finally, a discussion like this can always lead into the "What are hit points, really?" discussion, and whether game damage is proportional to energy impact in the first place. But even if hit points aren't raw structural strength, it's unclear whether damage should be increased or decreased in light of that, which is outside our current scope.

In summary: Steve Winter's "Kinetic energy is the key" is certainly a lot more correct than the Parker article -- but it would have been nice to acknowledge the effect of air resistance and know exactly how much of an approximation is being made for a game rule. By looking at velocity at the end of the fall, and computing the energy thereby transferred, in some sense we get a model that's in between the Parker and Winter models. That said, it's close to 95% Winter and 5% Parker in their proposals.