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I am puzzled by a riddle to which I have been told the answer and I have loads of difficulties to believe in the result.

The riddle goes as follows:

"imagine you are shrunk to the size of a coin (i.e. you are, say, scaled down by two orders of magnitude) but your density remains the same. You are put into a blender of hight 20 cm. The blender will start working in 60 seconds, what to you do? "

One of the best answers is apparently: " I jump out of the blender to escape (yes the blender is still open luckily)"

This seems ultra non intuitive to me and I have tried to find flaws in this answer but it seems to be fairly robust.

There are two ways you can think of it:

  • the mass scales as $\sim \: L^3$ and therefore it will be $10^6$ times smaller. If we imagine that the takeoff velocity $v_{toff}$ is the same as before being rescaled $v_{big}$. We get then the hight at which a mini us can jump by equating the takeoff kinetic energy and the potential energy i.e. $mv_{toff}^2/2=mgh $ $\Rightarrow$ $h_{mini} = v^2_{big}/(2g) = h_{big} \sim 20 \rm cm$

  • The second way to see it is to look more in details on how the power produced by muscles scales with the size of muscles. Basically, the power scales with the cross section of the muscle i.e. with the number of parallel "strings" pulling on the joints to contract the muscle. This implies that $P_{mini}=P_{big}/\alpha^2$ ($\alpha$ being the factor bigger than 1 by which you have been rescaled). We know that the takeoff kinetic energy will be given by $P \Delta t$. We assume now that $\Delta t \sim L/v_{big}$ so that $\Delta t_{mini} = L/(\alpha v_{big})$. In the end, this calculation tells us that $E_{mini} \sim E_{big}/\alpha^3$. However, equating again with the potential energy to get the hight we have $h_{mini} \sim E_{mini}/(m_{mini}g) = (E_{big}/\alpha^3)/(gm_{big}/\alpha^3)=E_{big}/(m_{big}g) = h_{big} \sim 20\:\rm cm$

These two reasonings seem fair enough and yet I don't trust the result they lead to. I would like to know if I am experiencing pure denial because of my prejudices or if there is some kind of flaw in the reasonings above (e.g. the fact that it is always assumed that the speed is unchanged when changing scale).

I also know that some tiny animals can jump more or less as high as human beings but it seems that most of the time these species have to use some kind of "trick" to store elastic energy in their body so as to generate enough kinetic energy at the takeoff to effectively jump super high.

If anyone has any thought on this, that would be very much welcome.

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I would rather say that $\Delta t\propto L/v$, but this does not change the answer $v_{\mathrm{mini}}=v_{\mathrm{big}}$. –  Peter Kravchuk May 30 '13 at 12:11
    
Shouldn't your mini muscles be able to produce 100 time more force (proportionally), ie allow you to accelerate significantly more. This would be supported by the fact that your legs and muscles would be much larger than most other creatures that size. Except maybe fleas –  Jim May 30 '13 at 16:22
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also, if you weigh only 0.1 g or less, how hard would if be to jump just before the blades start up and ride the wind out? –  Jim May 30 '13 at 16:24
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The problem is undefined because the scaling of the muscle power, or takeoff speed is not specified and you have to speculate to get an answer. There are a million ways you can scale muscle power or take-off speed. –  ja72 May 30 '13 at 18:56
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@innisfree: I agree that it is an experimental fact that ants can carry and pull very heavy loads when compared to their weight and size, I don't deny that. I am just suspicious when it comes to the scaling explanation from physicists. Afterall, some humans are able to lift cars and pull aeroplanes with their mouth! Also, ants do not carry heavy weight of their shoulders but on their back which allows to distribute more evenly the load they have to carry. Not to mention that to pull something heavy [to be continued] –  gatsu May 31 '13 at 11:24

3 Answers 3

Think about it like this: in order to get all lengths (i.e. your own length as well as the height of your jump) to scale down by a factor $\alpha$, while keeping the contraction velocity of your muscles the same, you have to rescale all lengths and all times occurring in the problem. That means that you have to scale down the gravity (length over time squared) by a factor $\alpha/\alpha^2= 1/\alpha$.

In other words: if you can jump up half your own heigth, after a scaling down in linear size by a factor 100, you would still be able to jump half your height on a planet with 100 times the gravitational acceleration of earth.

So effectively what is assumed is that you can arrive at a consistent physics when you scale down all length and all time scales by a factor $\alpha$, and your mass by a factor $\alpha^{-3}$. The dimensionless jump height (jump height divided by body height) remains invariant under this scaling down operation. However, the point is that in this scenario you also have to rescale the gravitational acceleration (length over time^2) by a factor $\alpha^{-1}$. It goes without saying that if you don't do this and keep the gravitational acceleration as weak as it is before scaling, you will be capable to jump much higher in terms of your own body height.

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The point is that I don't really "want" anything appart from knowing if I am being close minded not believing in the answer suggested. It is true that I intuitively imagine that what will be conserved is more the fraction of my height that I can jump more than the actual velocity. As ja72 pointed out this is something assumed in the two reasoning presented and I do not know if it is a valid argument or not. –  gatsu May 30 '13 at 19:35
    
Added a paragraph to my answer to clarify the assumption common to both reasonings. –  Johannes May 31 '13 at 2:14
    
Thanks Johannes. I knew this reasoning that tends to say, "well, if you are scaled down 10 times then you will weight 1000 times less and it is equivalent to keep you the same size and dividing by 1000 the gravity". This argument still implicitely assumes that the takeoff velocity will be unchanged upon being shrunk. But is it true? Are we dealing with such simple physics here? –  gatsu May 31 '13 at 11:10
    
You have to specify what you mean by "but is it true"? The scenario itself is purely hypothetical and therefore not "true" (we don't have the technology to shrink a human), but on the other hand the scaling behavior (length and time scales scaling by a factor 0.01, mass density remaining invariant, muscle strength scaling by a factor 0.0001, muscle work performed by a factor 0.000001, etc.) is physically and physiologically consistent. –  Johannes May 31 '13 at 16:40

Let's assume our gravitational potential is zero at our center of mass just before the jump. Our initial mechanical energy is zero. We do nonconservative work to increase our mechanical energy. Then our feet leave the floor and our kinetic energy diminishes until we reach height $h$. We have $$W_{\mathrm{nc}} = F d = \frac{1}{2}m v^2 + m g d = m g h.$$ Therefore, $$h = \frac{F d}{m g}.$$

Strength is proportional to area and mass to volume so relative strength is inversely proportional to length, $F/m = k/L$. (Take $L$ to be our height, for example.) In addition, the distance we travel before leaving the ground is proportional to $L$, $d = k' L$. Therefore, $$h = \frac{k/L \times k'L}{g} = \mathrm{const}.$$ Thus, we will jump just as high as we had jumped before we were shrunk! Since a blender sized jump is a modest one for a normal man, we should be able to jump out without too much difficulty.


Addendum: Here's a simple argument why power scales like force. For constant acceleration and zero initial velocity $v^2 = 2 a d$. Therefore, $F = m a = m v^2/(2 d)$ and so $$v = \sqrt{\frac{2 F d}{m}} \propto \sqrt{\frac{L^2\times L}{L^3}} = 1.$$ Therefore, instantaneous power scales like force, $$P = F v \propto L^2\times 1 = L^2.$$

A key fact regarding the scaling of force that may not be apparent is that we assume muscle fibers have a size that does not scale with $L$. (This is certainly true of humans of different heights and should make intuitive sense, smaller animals aren't made out of smaller molecules.) Thus, a leg with a smaller cross section has fewer fibers---the area of the cross section essentially counts the fibers.

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It is difficult for me to know which is the one that is simply scaled done with respect the cross section: force or power? There is a difference between the two as talking about force won't need any time scale involved while talking about power will...and that is precisely the only flaw I see in these reasonings. The issue of transporting oxygen into the muscles seems to give a time scale for instance. –  gatsu May 31 '13 at 11:34
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Oxygen transport is not a limiting factor for the impulsive (jump) power delivered by a muscle. –  Johannes May 31 '13 at 16:57
    
So, conversely, if we scale up a fly to the size of a human, it would not be ultra strong? In other words, Seth Brundle did not really break the other arm-wrestler's arm? –  Eugene Seidel Jun 1 '13 at 13:06
    
@EugeneSeidel: It depends whether the fly has a higher strength to weight ratio then our shrunken man ... scaling arguments don't capture the fact that a fly and human have a completely different morphology. :) –  user26872 Jun 1 '13 at 17:39

The scaling arguments shown above are correct, which is consistent with the fact that animals of very different sizes - from a flea to an elephant - jump to the height on the order of 1 meter. Perhaps insects are different from mammals; so let's say a mouse vs. an elephant is a legitimate example of scaling (by about 100 in linear dimension) of a pretty similar organism - but both jump to about same height.

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I did not know that an elephant could jump about 20 cm. Moreover, people tend to give examples of particular small animals that can jump as high as humans but it is difficult to assess if they are exceptions or an actual trend in the animal kingdom. I have difficulties to imagine a mouse jump 20 cm for instance. –  gatsu May 31 '13 at 10:59
    
elephants can't jump –  Jim May 31 '13 at 13:38
    
@gatsu: Mice can jump at least 12 inches, as is well known to the Department of Health of New York State. Elephants are poor jumpers because their strength to bodyweight ratio is so low. Relative strength scales as $1/L$. If we were to grow instead of shrink, at some point our relative strength would be so low that we could not jump. At some point our bones would be crushed under our own weight! –  user26872 Jun 1 '13 at 3:40

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