Supersonic Man

November 13, 2011

the square-cube law: “All animals jump the same height”?

Filed under: Hobbyism and Nerdry — Supersonic Man @ 6:38 pm

It was a school-age best friend’s dad — a physicist at the Stanford Linear Accelerator — who proposed the hypothesis “All animals jump the same height”.  Now this did not of course mean that a hippo can jump the same height as a gazelle… what it means is “It seems like the ability to do a standing leap to a given height is fairly independent of the size of an animal’s body.”  Humans and cats leap a similar upward distance, and so do grasshoppers.

So the question is, if you apply the square-cube law and the like to the issue of jumping, does scale cancel out?  Or does it only approximately do so?  Is size an advantage in jumping ability, or a disadvantage, or neither?

This turns out to have a lot of complications.  Muscle force scales with cross-sectional area as the square of size, while muscle power scales with volume as the cube.  The distance through which the muscle does work scales with the first power, while the time over which it’s exerted may scale differently, I’m not sure yet.  The weight being launched scales as the cube.  Bone thickness as a proportion of limb thickness tends to scales as some unknown low power of size… but maybe that’s not relevant.  Some quite large animals, such as a mountain lion, can have bodies where a strong majority of their total weight is muscle, and some bugs have little muscle by weight because many of them have lifestyles that never call for much strength.

Here’s one variable we can probably solve for: the duration of the work done during the jump is limited by the muscle’s power capacity.  But it’ll also be limited by force.

If we go by force, then doubling the scale in each dimension means we have four times the force acting against eight times the mass, therefore half the acceleration.  (Of which gravity subtracts a fixed amount — this may be where things can’t cancel out.)  Gravity aside, stretch this over twice the distance of lift, and by t² = 2d/a,  you get twice the time.  So four times the force over twice the distance makes eight times the work, but eight times the work over twice the time makes only four times the power.  So there’s a capacity going unused there… maybe it means that as you scale an animal up, it can use more leverage — longer legs, if you will — to gain extra leaping ability.

The crucial number is how fast it’s moving when the feet leave the ground.  Half the acceleration for twice the time makes the same final speed.  But only in the absence of gravity.  So on the one hand, the larger animal pays a greater penalty to gravity in ability to accelerate, and therefore loses some hang time relative to the small animal which gets up to speed more quickly.  But on the other hand, it’s gained more altitude during the leap before its feet leave the ground, which has to count as part of the total jump height.  And what about this possible longer-lever factor?

Let L be the lengthening of the leg for leverage.  Force exerted on the ground, and therefore acceleration, decreases by a factor of L, while distance that force is exerted through increases by L.  As above, this means that the time of acceleration also increases by L.  So the ending velocity is the same… except that the gravity penalty increases.  Total work done stays the same, but the time increases, so the power output is adjusted by 1/L… it goes down.  Hm — it turns out that a larger animal is better off using a shorter leg, not a longer one!  How much shorter?  Crap, it looks like L should be the reciprocal of the scale factor… but that’s nonsense, because it means that a spider and a lion should have legs of the same length!  Clearly something has been misinterpreted here.

We could compromise and say that L is the square root of the reciprocal of the scaling factor.  So if one animal is four times as big each way as another, and 64 times as heavy, then its legs are only twice as long.  That seems to roughly fit nature if you look at some cases, like daddy-longlegs vs pigs… but there are plenty of counterexamples.  For instance, mice have far shorter legs relative to body size than deer.  Doesn’t seem to slow them down much.  I’m thinking that leg length is just something that nature optimizes by other criteria outside the scope of this question, and there’s not much we can say about it.  It may be that it ends up having less real effect than you would assume.  So let’s drop the leg length issue.  From what I can intuit by watching real animals, it seems to make a lot less difference than one might suppose.

To deal with the gravity issue, let’s try looking at some concrete magnitudes to bring into the equation.  How high do animals actually jump?  The maximum height appears to be somewhere around one meter.  For most animals, it’s less; in humans, for instance, only top athletes get that much air.

If a flea-sized speck could launch itself upward by one meter, its takeoff velocity would have to be 4.43 m/s.  (Which would be lost to air resistance faster than to gravity, so it wouldn’t actually reach that height… but we can ignore that issue for big animals.)  If a larger animal accelerates through 20 cm, it needs a takeoff velocity of 3.96 m/s to cover the remaining 80 cm, which requires an acceleration of 39.2 m/s².  That’s four gees, so the gravity penalty is a 25% increase in power and force required.

Now scale the animal up by two, say from cat sized to dog sized: eight times the weight and twice the acceleration height.  Only 60 cm left to cover after takeoff, so the speed is 3.43 m/s.  The acceleration required is only 14.7 m/s², a gee and a half, so the gravity penalty increases from 25% to 67%.

So who comes out ahead, the littler animal or the bigger animal?  For each kg of body weight, the small animal has to exert an upward force of 49 N to gain that speed while also overcoming gravity.  The large animal has to exert only 24.5 N/kg (for twice as long a duration), but it has eight times as many kg.  It has four times the force-exerting capacity, or one half the capacity per kilogram.  Well look at that, 24.5 happens to be exactly half of 49!  It came out dead even in this test case — the change in demand exactly matched the change in capacity.  Neither animal has any advantage of being able to make the jump more easily than the other.

I haven’t worked out the math for a general case yet, but it looks like within a certain domain, at least, things cancel out pretty darn well.  (Of course these numbers also imply that above a certain size, animals can’t jump at all.  That’s not unrealistic.)

…But that leaves out the issue of power capacity.  The two animals are doing the same amount of work per kilogram (the larger one does half the force for twice the time), but the large one has a lower power output, because of that same increase of the duration of the acceleration phase.  So in that sense the larger animal has an easier time.  It may be that an animal below a certain size is unable to keep up with bigger ones because the muscle is not able to put out the necessary work in a brief enough span of time.

We can note that fleas use a workaround for this problem: they exert muscle force over a longer interval to compress a spring — a pad of rubberlike protein in one of the bug’s back leg joints — which then releases the energy all at once for the actual jump.  Critters that tiny probably can’t jump very well without using such a technique.

[update] Here is part 2, in which I find that there really is a mathematical identity across scales.


  1. “It was a school-age best friend’s dad…”

    Sadly, he just passed away.

    Comment by Supersonic Man — January 1, 2014 @ 11:22 am | Reply

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