What's wrong with Arnold's scaling argument on jumping height?

The following question was put on hold: Is it possible to prove that an elephant and a human could jump to the same height?

It reminded me of an exercise (24a) on that exact topic in Arnold's "Mathematical Methods of Classical Mechanics". The solution he gives goes like that:

For a jump of height h one needs energy proportional to $L^3h$, and the work accomplished by muscular strength $F$ is proportional to $FL$. The force $F$ is proportional to $L^2$ (since the strength of the bones is proportional to their section). Therefore, $L^3h$~$L^2L$, i.e. the height of a jump does not depend on the size of the animal. In fact, a jerboa and a kangaroo can jump to approximately the same height.

The comments of the above question tended to dismiss that argument. What's wrong with it?

Addendum: It seems obvious that not all animals jump exactly to the same height, given their different physiologies/shapes. Some of them can't jump at all.

The question is to be understood in the following spirit: if we plotted jumping height vs animal size for a lot of different species, would there be a correlation? I don't mean there is no spread; I totally expect a big spread due to the other factors involved.

Second addendum: Some interesting points have been raised in comments and answers. I will accept an answer that incorporates: the domain of validity of Arnold's argument (or explain why it is never valid), the effect of air drag on very small jumpers and the impossibility of very large animals.

Bonus points for documenting the yet elusive jumping elephant and plotting jumping height vs size for different species ;)

• More on animals & scaling: physics.stackexchange.com/q/10793/2451 , physics.stackexchange.com/q/72641/2451 , physics.stackexchange.com/q/153538/2451 and links therein. – Qmechanic Mar 13 '18 at 10:05
• The argument is perfectly sound and valid. Remember however that by definition scaling calculations only give you the functional dependence on key quantities, but not the numerical coefficients. Similar arguments can be found in the book The Physics of Superheroes by J. Kakalios, when the jumping height of Spiderman is compared to that of a normal spider. – valerio Mar 13 '18 at 11:55
• Relevant reference: web.mit.edu/6.055/old/S2009/notes/jump-heights.pdf – valerio Mar 13 '18 at 12:24
• I wouldn't go so far as to say that the argument is perfectly sound and valid. The argument is simplified physics, not biomechanics. How realistic and applicable it is depends on a vast number of things that are unlisted yet implicated assumptions when one is doing a scaling argument. For example, we are assuming muscle quality is the same for different animals, or that the leverage mechanisms are the same/similar. – HsMjstyMstdn Mar 13 '18 at 14:24
• @valerio While I understand the gist of your comment, I wouldn't agree that you get the correct functional dependence as the powers that the quantities are raised to are usually not the integer ones you see in scaling arguments. I would also add that the scaling argument is a very rough approximation that doesn't hold well at all across animal size scales. These are pedantries, but I think they are important ones given OP's question. – HsMjstyMstdn Mar 13 '18 at 14:53

There is nothing wrong with the argument. The mathematics are quite simple and the conclusion is sound - scale cancels out.

Let's consider the essence of the question; How does the scale size of an animal affect the absolute height it can jump?

Let's assume an on-the-spot spring jump so we exclude a run-up. Now consider an arbitrary animal (let's call it a ballerina). It crouches, extends itself and leaps in the air. The difference in height of, say, its head between crouch position and when it leaves the ground is its leg extension ($s$). The difference in height between leaving the ground and top of the jump is the jump height ($h$).

As the ballerina jumps, its muscles provide an upwards acceleration of $a$ over the extension length, $s$ of its legs. This leads to an initial upwards velocity, $v$ that is given by: $$v^2 = 2as$$

As it flies upwards, it is decelerated by gravity until it stops at the top of its leap. The height reached is given by the same equation where the acceleration is that due to gravity, $g$: $$v^2 = 2gh$$ hence $$h = {v^2\over{2g}}$$

Since $v$ is the same in both equations, we can equate them, giving us: $$h = {{as}\over{g}}$$

Now, $a = {F \over m}$, so: $$h = {{Fs}\over{mg}}$$

We now have an equation for how high a ballerina can jump that depends on the force in its muscles, $F$, the length of its leg extension, $s$, its mass, $m$, and $g$.

What happens if we scale up the ballerina by a factor $x$? Leg extension simply goes up by $x$. Mass, which depends on its volume, goes up by $x^3$. Interestingly, muscle strength goes up by $x^2$ because it is the cross-sectional area of a muscle that gives it its strength (not its volume!). Plugging in the scale factors:

\begin{equation}\begin{aligned} h &= {{x^2 F \cdot xs}\over{x^3m \cdot g}} \\ &= {{{x^3}\over{x^3}} \cdot {{Fs}\over{mg}}} \\ &= {{Fs}\over{mg}} \end{aligned}\end{equation}

So we get the same height.

The conclusion is that the scale size of a creature is not a factor in calculating how high it can jump. To put it another way, the graph of jump-height versus scale size is a horizontal line.

Incidentally, elephants can actually jump - there is a circus stunt in which they stand on their two hind legs and hop slightly. They go up 10-20 cm, which is about the same as a flea.

• I would argue that there is something inherently wrong with the scaling argument. Specifically; it's too simplified and approximate. There are assumptions made and phenomenon ignored so that this specific result can be achieved. It's hard to find a signifigant direct coorelation between jump height and overall size. I don't think it's honest to conclude that jump height is therefore independent of size; when really it's quite dependant on size ratios and does vary for many animals – JMac Mar 13 '18 at 14:43
• @JMac I don't think anyone is disputing that some species can jump higher than others. The point is that the jump-height is not a function of size. If you take a random species (say, a rabbit) and find it can jump 30cm, then scale it up by a factor 5, it can still jump 30cm. This is the absolute height. Relatively speaking, the rabbit itself feels extremely leaden - it used to be able to jump its own height, now it can only jump a fifth of that! – Oscar Bravo Mar 13 '18 at 15:36
• Good answer. So many people here don't seem to get what a scaling estimate actually is, and expect this kind of calculations to give them the right numbers with $5$ sigma error, whereas they were never meant to do that...sigh! – valerio Mar 13 '18 at 22:04
• The argument is entirely bogus. An order of magnitude is a huge difference (and why just one order? it could well be 2 or 3 orders). I could guess the jump height of any animal up to 1-2 orders of magnitude without any reasoning at all. The argument about scaling the same animal is also bogus since there is no such thing as the same animal at significantly different scale. Even young of the same species look very different. If you scaled a rabbit to the size of an elephant it wouldn't jump 30cm, it would probably be unable to walk and maybe even live, crushed by its own weight. – Anton Fetisov Mar 14 '18 at 4:58
• @OscarBravo Just because the math is simple, it doesn't mean it's the right math. This assessment makes so many specific assumptions that I don't see how it has much physical value. You can come up with all sorts of correlations if you choose which factors to examine and how to constrain them in such a way that it fits the result you want to see. I'm really not convinced that isn't the case here. – JMac Mar 14 '18 at 11:59

What's wrong is :

For a jump of height h one needs energy proportional to $L^3/h$

Taking L as a measure of animal size then we should actually have

$$E \approx Mgh \propto L^3h$$

So not divided by $h$ but multiplied by $h$ !

And a little thought would show that dividing by $h$ would make no sense, as it implies you need less energy to jump higher. I don't know if that's a typo by the OP or in Arnold.

, and the work accomplished by muscular strength F is proportional to FL. The force F is proportional to $L^2$ (since the strength of the bones is proportional to their section). Therefore, $L^3/h \sim L^2L$

A serious flaw here is assuming all animals are built the same way. We're implicitly assuming that animals have both the same ways of storing energy for activities like jump and that the proportion of muscle mass available for jumping is the same, which is not the case.

Now I did a bit of back of the envelope modeling of this, as I think it's fair to say that the proportion of an elephant's volume that is leg is smaller than that of e.g. a human. Whatever the case we might modify the relationship to produce :

$$L^3h \sim FL \propto \frac {V_{legs}}{V_{total}}L^3$$

So at the very least we get

$$h \sim \frac {V_{legs}}{V_{total}}$$

Which casts a very different view of this idea that all animals jump to the same height. It now becomes a function of the design of the animal. It even tells me that other things being equal a fat me will not jump as high as a thin me, which sounds like a better model than we all jump the same height !

I see this is another answer :

all animals jump about the same height to within an order of magnitude - from about 20cm to 2m.

I really think of this as a cop-out argument. I've seen a similar argument in another place online and it boils down to "can't be bothered to work out a better model so ignore the order of magnitude difference". I just don't see the point of trying to model something if you're going to do that.

Finally

In fact, a jerboa and a kangaroo can jump to approximately the same height.

Do we know this to be true ? How do we even define the jump height for the case of two such different animals (which strikes me as a non-trivial issue) ? But is this another case of ignoring order of magnitude differences and calling it "the same" ?

And what if I replace "jerboa" with "elephant" - does it work out then ?

This sounds like very sloppy logic.

• I don't think your argument contradicts to Arnold's. Arnold is arguing the overall size has no correlation to jump height. Where your argument states relative sizes (body to leg): the value $V_{legs}/V_{total}$ can sink into a parameter has nothing to do with overall size ( such a parameter is related to the geometrical structure of the object though). – Shing Mar 13 '18 at 12:42
• It seems obvious that not all animals jump to the same height, given their different physiologies/shapes. The point is more that given one jumping mechanism, size doesn't matter; we could try plotting jumping height vs animal size for different species and see whether there is a correlation. I don't mean there is no spread; I totally expect a big spread due to the other factors. – Antoine Mar 13 '18 at 13:05
• @StephenG I hope you get the point that we are ignoring body design differences here. We are simply looking for the relationship between scale size and jump-height. The counter-intuitive conclusion is that there isn't one. – Oscar Bravo Mar 13 '18 at 14:02
• Note that "$L^3/h$" seems to have been a typo, and the present version (v4) of the question has $L^3h$ as you do. – rob Mar 13 '18 at 17:47
• @StephenG if you think "perhaps you should avoid physics entirely" is a reasonable response to a criticism of the tone of your writing, then, well, I will let the reader draw their own conclusion. – Nathaniel Mar 14 '18 at 3:42

The fact that animals are all very different aside, it ignores some important facts.

The first big problem is that it ignores the fact that an animal scaled up might not be able to stand at all. By the original argument, $F\propto x^2$ and $mg\propto x^3$, where $x$ is some scaling factor. So it should be obvious that, at some point, $mg>F$ and the animal is unable to stand, much less jump.

Connected to this issue is the definition of jump height that is used. It assumes that $mgh=Fs$, where $h$ is the height of the jump and $s$ is the extension of the legs. But this counts simply standing up as a "jump," even if the feet do not leave the ground. The proper expression would be:

$$mgh=Fs-mgs$$

subtracting standing up from the height reached. But this no longer scales the same way:

$$h=\frac{Fs-mgs}{mg}\sim\frac{x^2x-x^3x}{x^3}={x^3-x^4\over x^3}$$

So jump height is only approximately scale-invariant so long as the negative term in the numerator is negligible. Or in other words, only so long as $F\gg mgx$. Once this term is no longer negligible, jump height starts to decrease with scale.

An average human can leg press about twice their body weight once. So already $F\not\gg mg$. Scaling up a human only two times leaves them unable to stand under their own weight. A particular form for a biological organism is not remotely scale invariant- it only works at one particular scale.

In other words, all else being equal, an animal's jump height correlates inversely with its size. It is approximately constant only in the limit where an animal can already jump much higher than its own height. If there is not a negative correlation between size and jump height, it is because all else is not equal- the biochemistry involved is very non-trivial.

• Regarding your first big problem - OK, let's restrict the argument to the range below which animals collapse under their own weight. So no mile-high behemoths and let's keep to anything between a flea and an elephant. Your second point about jump height is a problem of definition; crouch-stand distance = leg extension, stand-top of jump = jump height. So no need for your additive term. Finally, about $F \gt mgx$; you are worrying about running into the collapsing range; Ok - so scale the other way - how high can Matt Damon jump in Downsizing? – Oscar Bravo Mar 14 '18 at 7:26
• @Oscar Bravo - On the contrary, I think this is an important addition to the argument, which is demonstrated in the answer to be relevant on the scales we are talking about, i.e. humans. I think you may be misunderstanding Chris's second point, which is basically the same issue as gravity drag in rocketry. As in your answer, the initial velocity is given as v² = as, and F = ma, but the F is the net force, not the applied force by the legs. Gravity is also acting, and so it needs to be accounted for. This is what Chris is doing. – brendan Mar 14 '18 at 14:43
• @OscarBravo The problem is that most animals are already not that far from collapsing under their own weight. You don't need mile-high behemoths- the size of an elephant is already much too big for the human form. Presumably because nature as a whole doesn't select for extra leg strength- it's just a waste of resources unless there's a specific reason to need to jump high. Matt Damon in Downsizing might be able to jump higher than before, but that doesn't necessarily say anything about an animal naturally that size. – Chris Mar 14 '18 at 15:35
• One might also notice that human high jumpers tend to be exceedingly tall. Elite high jumpers tend to be about 20 cm taller than average. – Chris Mar 14 '18 at 16:23
• @OscarBravo The net force does not scale with the square of the scale. It scales as $x^2-x^3$, since the downward force of gravity scales with the mass, not with the cross-section of the legs. This is irrelevant in the limit that animals can jump much higher than their own height, but most animals cannot, and so it is certainly relevant. – Chris Mar 15 '18 at 8:30

The force F is proportional to L2 (since the strength of the bones is proportional to their section).

That is relevant only if strength of the bones is the limiting factor. If an animal with height L jumps a height h, then the acceleration is gL/h. For a tick of height 1 mm to jump 1 m, it would have to have an acceleration of 1000g. A human with height 2 m jumping 1 m would have to have an acceleration of only 0.5g. For small animals, the larger they are, the higher they can jump, generally. The height they can jump increases more slowly than size, so smaller animals can jump a larger number of "body lengths" than larger animals. As animals get larger, the slope of the curve decreases, and there's a point at which it goes negative.

• When you say the higher they can jump, what do you mean by higher? Do you mean in absolute terms (i.e. more centimetres)? Or relative terms (i.e. a larger percentage of their body height)? Distinguishing between the two is crucial to understanding this question. – Oscar Bravo Mar 15 '18 at 8:28
• @Oscar I find this comment a bit odd, as absolute seems like the obvious default, and is what everyone else is talking about. Not only did I think I distinguished between the two reasonably clearly ("For small animals, the larger they are, the higher they can jump, generally... smaller animals can jump [a] larger number of "body lengths" than larger animals." ), AFAIK I'm the only one who tried to distinguish between them, so I don't see why you're commenting only on my answer. – Acccumulation Mar 15 '18 at 15:10
• Sorry - I re-read your answer. I misunderstood the sentence about body-lengths (there's a typo in it...). – Oscar Bravo Mar 15 '18 at 15:47

I've had second thoughts on the whole scale-invariance thing... Referring to my answer above, I think we're safe up to the equation for jump height: $$h = {{Fs}\over{mg}}$$ In the argument above, I re-scaled the variables and showed that the scale factor cancels out - hence jumping is scale-invariant. This means that if real Matt Damon can jump 30cm, his character in Downsizing can also jump 30cm.

However, as @Chris et al. pointed out to me, the force, $F$, in the equation above isn't just the force available in the muscles (which simply scales with the square of the scale-factor). Rather, it is the force used to accelerate the animal - and that force is the net force after gravity has been subtracted. In other words: $$F = F_{0} - mg$$ Now this force doesn't simply scale with size because it has $m$ in it... So the real jump height equation is: $$h = {{(F_{0}-mg)s}\over{mg}}$$ Now if we scale this by $x$, we get: $$h = {{(x^{2}F_{0}-x^{3}mg)xs}\over{x^{3}mg}}$$ which simplifies to: $$h = \left( {{F_0}\over{mg}} - x \right) s$$ Note that all variables refer to the values for the original animal (not the scaled version).

It is apparent that when $x=1$ (original situation), $F_0$ must be greater than $mg$ to get off the ground at all. This is obvious so that's good.

If we scale down ($x \ll 1$), the subtraction of $x$ is not significant and the scale-invariant behaviour is approximately obtained.

But, if $x$ increases, $h$ shrinks until, when $x = {{F_0}\over{mg}}$, $h$ tends to zero and jumping becomes impossible. This chimes more with reality.

• Thanks for the vote - even though it was -1 :-(... would you mind commenting on what you think is wrong? – Oscar Bravo Mar 20 '18 at 6:41

protected by Qmechanic♦Mar 13 '18 at 12:48

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