Monday, July 27, 2020

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 space or time. 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.


19 comments:

  1. I'd also conclude that that table is evidence in favor of a simple 10'=1d6 rule. It's within about 10% accuracy per 100' for the first 200', which should cover the majority of falls in my general experience.

    Accounting for air resistance isn't reasonably possible, since that's a function of mass, density, and shape. A sheet of paper and a crumpled ball of paper have the same mass/density, but one is subject to far more dynamic effects than the other. There are other further details being abstracted away already (e.g. did the PC suffer any extra trauma from landing on one surface vs. another?).

    ...and wow, I'm not sure if I've read that paragraph from Parker before, but it sure is a doozy 😂

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    1. I agree! Good point about the difference in weight shape; something more complicated would uncork issues of how we deal with hobbits, giants, etc.

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  2. https://www.npr.org/sections/health-shots/2018/08/24/641395468/surviving-a-big-fall

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    1. LOvely. Best quote: "Fortunately, we don't have enough data to make a trend line"

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  3. Converting this into a quick rule of thumb would look like this:

    * 1d6 per 10 feet fallen, up to 200 feet.
    * 1d6 per 30 feet fallen, from 200 to 500 feet.
    * 1d6 per 50 feet fallen, from 500 to 1,000 feet.
    * 1d6 per 400 feet fallen, from 1,000 to 5,000 feet.

    Pretty workable, especially given that falls of greater than 200 feet should be rare.

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  4. I'd be inclined to keep the d6-per-10' and add in a size modifier (something like "fall damage multiplied by x^2 where x is:
    1/2 for anything under 20 lbs (1/8th of 160)
    1 for anything size up to and including size M21 lbs
    2 for anything up to roughly horse sized (8 * 160 lbs)
    4 for anything beyond that

    Except, as pointed out above, a lot depends on win resistance and the surface being landed on. Even a crumpled dragon would generate a fair amount of drag with those wings. And it seems silly for a 20' fall to be potentially lethal to a hill giant. But maybe that's right?

    I dunno, this whole conversation sort of falls apart when we start seriously considering the biomechanics of an 18' tall human-proportioned biped.

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    1. Maybe a middle ground to adjust for the victim's size would be to measure by "equivalent heights" rather than absolute distance? It'd become uglier than just working in 10' increments, of course.

      In general, falls are more dangerous than people tend to expect. If I'm not mistaken, most industrial safety codes require protections (railings, tie-off points for harnesses, etc.) starting around 1.5'-3' potential fall distance, because that's enough for non-insignificant risk of serious injury or death. For that matter, there are plenty of cases of people just falling off of their feet and suffering bone fractures, concussions, organ damage, etc.

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    2. There's a Dragon article I think called, "How much does my giant weigh?" (or might be another later one) that winds up grappling with the issue that giant-sized humanoids aren't physically viable in first place. Winds up pointing to 1E cloud/storm giants with _levitate_ abilities, and ponders if other giants might have some pseudo-trait that makes it possible to walk? That's always stuck with me.

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    3. My impression has always been that it's their namesake "giant strength" that makes them viable. That they're just magically stronger and tougher, perhaps their very flesh and bone is denser and more resistant to stress. And that's also why a 12' hill giant is as hard to kill as a bull elephant, despite being several times less massive.

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  5. Interesting stuff! I am not a fan of falling damage in D&D, but 1d6 per 10 feet seems like a good compromise.
    If we apply this stuff as written we get strange results... 10th level characters survive 90%+ of 100 foot falls, where real survivability is less than 10% IIRC.

    But what is worse is that cats will die easily from a fall, while big quadrupeds probably wont, etc.

    Here is what I adopted in Dark Fantasy Basic, FWIW.

    Falling
    Falling causes 1d6 Constitution damage per 10 feet fallen (maximum 10d6). Roll a DC 20 Constitution and Dexterity save. If one succeeds, the damage is halved; if both succeed
    it is divided by three.

    Creatures and Constitution
    If a creature has no Constitution score, treat them as if they
    had Constitution 10. Most creatures are not necessarily
    better or worse than people at surviving starvation,
    dehydration, falling, etc.
    If weight is an issue (or if the GM wants giants to survive a
    fall form their own height), creatures of different sizes take
    HP damage instead of Constitution damage. It will take lots
    of poison to kill an elephant, but not a cat, for example.

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    1. Neat. It's definitely something I grappled with (via spreadsheets). I think I've become okay that 10th-level PCs, like Conan, really have some supernatural survivability (there's a link above on some people surviving 10K+ feet falls; the 10th+ PCs are they?).

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  6. I'm not sure energy should be the sole calculation. I don't think the body reacts equally as energy increases.

    Under this chart a fall from 40' to 50' increases the amount of damage received by 25% (from 4 to 5). Yet the odds of dying for that minimal 25% increase should go way up. It likely shouldn't be just an energy calculation, but a "what effect does that increase in energy have in survivability.

    Your percent collision model graph from your earlier article reflects this. Fatality doesn't map directly to energy.

    (I'm confused by your statement that you've not accounted for air resistance - it's in your Feb. 2018 article?)

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    1. Good spot. But the only thing my 2018 article is that I was ignoring air resistance for short falls, and recognizing a terminal velocity from it on the other end (and nothing on the curve in between, which is what I'm adding here).

      I'm not sure the odds of dying jump up so much between 40' to 50'? There's a chart of data points in a Dickinson-Roberts article, but I haven't had time to transcribe/assess it to date.

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  7. At the core is, "how scary is falling to a PC". If a 50' fall is 5d6 (avg 18 hp, max 30') most mid-level PCs will be willing to give it a go (or even plan to shrug it off).

    Most humans won't be willing to give it a go. You're using physics to determine the die to use, but using dice at all isn't physics.

    Modeling from reality backwards provides a more "realistic" model. As long as you're using just dice.

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    1. Well, it would be scarier if they were crippled for their natural pc life lol.

      https://aab-edu.net/

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  8. Glad d6/10' still "fits". I jsut get a kick out of the Time column. Depending on how long of a round one uses, you are falling quite far.

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    1. Come to think of it, this sort of highlights how ridiculous Feather Fall is. If any given spell has a casting time of 1 full round (with FF being the common exception to this), then a M-U wouldn't have time to cast FF unless falling a distance of over 1000 feet!

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  9. The reason why a giant would take more damage than a human is that the Force increases cubically (F=m*a, m=volume*density, volume=side³) but the collision is made in an area (side²)... So as you increase the size of the object, you increase the impact force cubically but the impact area squarely.
    Another issue is that most animals are just a bag of water so when they fall and splash on the floor, there is a lot of dynamics physics about fluids and deformation and calculus that is better not to trigger some PTSD for folks like us that graduated in STEM. Hehhehe
    In a world full of magic, 1d6 per 10ft seems a fair compromise backed by sufficient data your provided.

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    1. LOL @ "better not to trigger some PTSD for folks like us that graduated in STEM"

      Agreed with all of that, well put.

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