# a man is broken, a horse splashes.

1. Mar 10, 2012

### superalias

"... a man is broken, a horse splashes."

"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."
— J.B.S. Haldane, biologist

No surprise there, given that the bigger the creature, the more momentum and kinetic energy it brings to the fall. (I believe the effect should hold even without considering terminal velocity, which in such a long fall is also worse for the big creatures.)

Also, biologists often mention the "square-cube law": if a creature is scaled up in all three directions by some factor y, it'll have y^3 times the mass but only y^2 the "structural strength" (for load-bearing and other purposes). So in a fall, maybe we'd expect the bigger version creature in the fall to come off "y times worse"?

Whatever "y times worse" means in terms of actual effect, that is. I'm trying to describe the outcome in more detail than just "the horse has more momentum and kinetic energy" or "well, the square-cube thing kicks in". But when I try to more precisely visualize what happens, I run into difficulty. If you'd be kind enough to humor me in visualizing the gory detail:

For simplicity, imagine two cube creatures made of generic fleshy stuff. One is a cube 1cm on each side, with area on each side of 1cm^2, and a mass of 1 gram. The other is identical except it's scaled up: a cube 10cm on each side, with area on each side of 100cm^2, and a mass of 1000 grams.

They both strike a hard surface at equal velocity, each flatly on its bottom surface. Say that, by some measurement, the smaller cube suffers injurious crushing and deformation (ruptured vessels, small broken bones, etc.) up to a depth of 0.2 cm; above that, it's relatively unaffected.

Now we also look at the big cube, which should come out even worse, right? Well, here's a take that more or less restates the "square-cube" viewpoint:

The big cube strikes with 1000 times the mass (and thus momentum and KE). But the impact is spread out over 100 times the surface area. So relative to the 1cm^2 small cube, each cm^2 of the big cube's bottom surface has 10g of mass above it, collides with 10 times the momentum and KE, and is subject to 10 times the pressure.

And... now what? What might that tell us about final effects? If the small cube suffers serious crushing to a depth of 0.2cm, might we expect the big cube to suffer similar crushing to a depth of 2cm? That's what jumps to mind – but if it's true, it seems the big cube is no worse off, as both suffer crushing to the same relative depth, 1/5 linear dimension.

To come out worse, the large cube would have to suffer more than 10 times the depth of crushing. Is there a plausible model that would yield that effect? For example, I've seen some reports claiming that projectiles fired into clay tend to penetrate to a depth that scales with projectile momentum. If that's true and if the cubes could be modeled as such, using the above crushing depth as a proxy for penetration, then still we'd have the big cube suffering only 10 times the crushing depth of the small cube.

Maybe springs are a better model. Assuming Hooke's Law holds, the maximum compression distance should be proportionate to m/k (mass / spring constant)... so again, the big cube's compression is 10 times as much for each cm^2 of surface, meaning no relative difference...

Or wait, maybe this is the answer: If the big cube is made of the same stuff as the small, shouldn't its k vary with dimension? If the big cube's k is only 1/10 the small cube's k, then the big cube would suffer 100 times, not 10 times, the compression. (That extends beyond its height, so the expected "splat" may be in order...)

If I'm on the right track there, then maybe the spring model is good. But then again... if the cubes are experiencing permanent crushing deformation, wouldn't that mean that the classic Hooke's Law spring model is necessarily the *wrong* model?

I know collisions are complex, with no simple plug-in formulae for effects like deformation, and imagining creatures makes it more uncertain. I'm not expecting any precise answer; a crude model, involving really simplified creatures, is fine. Is there some model and some set of conditions, whether numerical or just descriptive, that's at least ballpark-level instructive for showing just how my hypothetical big cube creature comes out worse in a fall?

2. Mar 10, 2012

### DaveC426913

Re: "... a man is broken, a horse splashes."

No. Let's take the best case. The critter lands on its feet.

The small cube has legs that are able to withstand the fall. The pressure on its leg bones does not exceed the compressive strength of generic bone material.

But take that same generic bone material, expand its area 100x (so we're looking at the big cube now) and it will withstand 100 times the pressure.

But it's not experiencing 100x the pressure - it's experiencing 1000x the pressure i.e. per unit area, the larger creature's bones exceed the compressive strength of generic bone material by something less than 10x. So its bones break.

3. Mar 11, 2012

### cng99

Re: "... a man is broken, a horse splashes."

Nicely explained, Dave

4. Mar 11, 2012

### superalias

Re: "... a man is broken, a horse splashes."

Thank you for the reply!

It may be a silly picture, but I was envisioning limbless cubes as the simplest example, avoiding any complications involving falling on limbs or not doing so. I should have made that clear, and not mentioned bones at all, sorry… but, perhaps it shouldn't matter; even if the critter is a limbless, boneless cube, and the injury in question is mainly rupturing of organs, your explanation should hold, I believe.

It's the "square-cube" effect I mentioned: 100x the resistance to pressure, but 1000x times the pressure, resulting in 10x the pressure per unit area – and thus, as you note, pressure in excess of what the poor organs (or whatever breaks in there) can withstand. Sounds good!

What I'm trying to get at, though, is just a step further:

Deformation won't be even throughout the cube; it'll be more severe at the bottom of the cube, less so or none at the top.

Assuming we can set some measurement for how "far up" the permanent (should I say plastic?) deformation occurred in a given cube, is there any guideline for how "far up" it will appear in another cube, all factors being equal except cube dimensions?

Maybe there's no answer for odd flesh cube creatures, but how about better-known materials? For example, if a given fall notably deforms the bottom of a 1cm^3 clay cube to a "height" of y millimeters, is there a model predicting the "height" of deformation in a 10cm^3 clay cube? Or for some other materials?

(And whether there are such models or not, am I right in guessing that using displacement of differently-dimensioned dropped springs, behaving per Hooke's Law, is a poor starting point for a model, since I'm interested in permanent deformation?)

If my question is inherently unknowable without vastly more info, or just inherently too goofy, please say so. If nothing else, a pointer to what laws and properties I should learn more about would be a big help.