# Why do smaller animals survive falls from larger heights?

I remember thinking about this classic problem with my friends back in school and tried to work through it again from a physics point of view. It turned out that it is more complicated that I thought:

Why do smaller animals survive falls from larger heights?

I would specifically like to know how physical scaling laws affect the chance of survival of an animal, i.e. we suppose the animals can be modeled as the same shape but every length scaled by a factor $k$. There are two biological factors one has to assume, namely that animals have roughly equal density and the force a muscle can apply scales according to the cross-sectional area it has ($\propto k^2$). I think apart from that one should be able to get away with making reasonable physics approximations. So maybe to reformulate what I mean by the "why" in the question above: can the phenomenon that smaller animals survive falls from larger heights be explained by physical scaling laws?

• Your question sounds like you take it for granted that each animal (species) has a characteristic "maximal falling height" above which it will die, under which it will survive, and that this height decreases with animal size. This is not evident at all. E.g. there are cases of humans accidentally falling from sky without parachute and surviving. I also remember a science TV show citing a statistical study of wounds on cats fallen from building wrt floor number: wounds increasing from 1st to 7th floor, stagnating to 11th, then decreasing over 11th! – L. Levrel Apr 11 '16 at 13:53
• Another version if the same question is 'why do elephants have such thick legs compared to spiders?'. The answer to both is the same: surface to volume. – tfb Apr 11 '16 at 22:59

Let me try. Let's take an elastic ball (instead of an animal...) of radius $R$ and stiffness $k$, freely falling in air. The ball breakes if compressed beyond a critical fraction of its radius. Let's wait it to reach its terminal speed $v_\infty \sim \sqrt{R}$. Hitting the ground, it will be compressed by an amount $x=v_\infty\sqrt{m/k} \sim R^2$. So the compression scales quadratically with radius, and small balls have more chances to "survive" the shock than big balls.

• does this actually take into account the muscle scaling? I assume it's hidden in the modelling as a spring, but not quite obvious to me – Wolpertinger Apr 12 '16 at 10:10
• Thank you for accepting the answer, but actually, now that I think, considering a constant stiffness $k$ is probably a bad approximation. If the ball/animal is just sitting on a hard surface, it is probably more reasonable to think that its body structure is such that it is compressed by its weight by a small amount $x_o$ independent of its size. So to have $k x_o=mg$, the bigger the stiffer, not to break under its own weight. But then the compression due to the shock becomes $x \sim \sqrt{R}$, and all becomes less obvious to me... – scrx2 Apr 13 '16 at 20:33

I can't claim to know the full answer, but it's amusing to note that this question has a fairly long history in biology. Way back in 1928, J.B.S. Haldane (my great-great uncle) wrote a popular-science article called "On being the right size", about the importance of scaling laws for biological anatomy. The following passage is relevant for the question at hand:

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, provided that the ground is fairly soft. 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. Divide an animal’s length, breadth, and height each by ten; its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistance to falling in the case of the small animal is relatively ten times greater than the driving force.

Haldane claimed (without offering direct evidence) that the difference could be explained because of air resistance. The force of air resistance scales as $f_r \propto L^2$, where $L$ is a characteristic linear dimension of the animal. However, the force due to gravity scales as $f_g\propto L^3$ (assuming fixed density). The ratio of these quantities scales as $$\frac{f_r}{f_g} \propto L^{-1},$$ meaning that smaller animals are buoyed up more effectively by air resistance as they fall.

I expect that there is probably more to the story than this, however, as noted in the comments, these sorts of experiments are tricky to get funding for...

One of the reason is the lack of "scale invariance", first noticed by Galileo in his "Two New Science". Galileo argued that animals cannot simply be scaled up, since their scaled bones would not support their scaled masses. For instance, an animal twice as wide, thick and tall, would be eight times heavier. Their bones would have to support eight times more weight. But their cross section is only four times larger. The scaled bone would not support the animal.

Consider for instance a small monkey and a gorilla. With some approximation we can consider them the same scaled animal. Let us say the gorilla is twice taller, thicker and wider. Then the gorilla's bones would be only four times thicker than the small monkey's one. This means only four times more resistant. If both animals fall from the same height, in the same way, the impact force on the gorilla would be eight times larger. As long as their bones have similar composition (which I assume it's true) the gorilla's bones would break before.

Another example, given by Feynman: People make those incredible cathedral and castles out of matchsticks. But we don't see any scaled real building like those.

Of course this is not the complete story. From the biological viewpoint there must be other regards. I'm just doing what physicists do, simplifying as much as possible the problems to find some approximated answer.

• thank you for your answer! could you elaborate how this specifically applies to falling? also I think the muscle scaling law i provided in the question accounts for that unless we go to extremes. I do realize that it's probably hard to compare an ant and an elephant by a scaling law, but I think with say a cat, a monkey and a human we could definitely get somewhere! – Wolpertinger Apr 10 '16 at 19:25
• I edited my answer trying to elaborate it. – Diracology Apr 10 '16 at 19:58
• thanks for your effort! this is two of the effects that would indeed provide an explanation in the argument you gave. But for example if you account for what Mark Mitchison said about air resistance, it amplifies the effect from the muscle/bone scaling. I did some calculations myself and I found 2 further effects which work in certain approximations that completely cancel the others. So I'm afraid what I am really asking, in particular for the bounty, is "the full story". – Wolpertinger Apr 10 '16 at 20:04
• Certainly the air resistance would be relevant in comparing the falls of a small frog and a monkey. But it would not be relevant comparing the 10 meters falls of a chimp and a gorilla. – Diracology Apr 10 '16 at 20:09
• even if we make that approximation (which is not a very good one I think, terminal velocity would probably be a better approximation) you are still missing at least one effect, which is what I call "fractional compressability" (in lack of a better word): to state it intuitively if a mouse gets squashed by 5cm its literally gone while a human would probably only break a couple of bones. We measure velocity (and hence acceleration) wrt distances though and not compared to the animals actual length. I.e. you have to account for a smaller allowed deceleration distance on impact. – Wolpertinger Apr 10 '16 at 20:16