Shrunk and thrown into a blender

[tommy01]

Hi everybody,

I encountered this question as an example of what a google applicant is asked:

You are shrunk to the height of a 2p coin and thrown into a blender. Your mass is reduced so that your density is the same as usual. The blades start moving in 60 seconds. What do you do?

It is cited in various media, e.g. in wired and they all give the following answer:

Those who were paying attention in rocket-science class will recall the formula for the energy of a projectile: E = mgh. E is energy (of a bottle rocket, let's say), m is its mass, g is the acceleration of gravity, and h is the height the bottle rocket attains. The height increases in direct proportion with energy (as long as mass stays the same). Suppose you tape two bottle rockets together and light them simultaneously. Will the double rocket go any higher? No; it's got twice the fuel energy but also twice the mass to lift against gravity. That leaves the height, h, unchanged. The same principle applies to shrunken humans jumping. As long as muscle energy and mass shrink in proportion, jump height should stay the same.

This argument seems to be wrong to me, for it would also apply to every insect of that size. "Muscle energy" is a somehow strange concept here ...

Any answers are appreciated.

Kind regards,
Tommy

 

[Drakkith]

Since legs are levers, not rocket engines, I don't see the given explanation as accurate. But I don't know for sure.

 

[A.T.]

Assuming uniform scaling:
Body mass is proportional to bodyheight³
Muscle force is proportional to bodyheight²
Distance over which the body is accelerated is proportional to bodyheight

 

[voko]

"As long as muscle energy and mass shrink in proportion" is an assumption, whose validity was not justified.

 

[AlephZero]

"As long as muscle energy and mass shrink in proportion" is an assumption, whose validity was not justified.

If you want a practical demonstration of that, fleas can jump 100 times their own height.

 

[jbriggs444]

Other complications:

A scaled-down launch will be quicker than normal. The muscles will need to build to full force more quickly. This effect works to reduce the height of a scaled-down jump. However, it is plausible that a scaled down jumper would be able to do all things more quickly (e.g. shorter nerve fiber runs).

A normal human jump involves pulling up the feet and otherwise contorting the body to clear obstacles without necessarily raising the center-of-gravity a corresponding distance. With a smaller body, the ability to do this will be reduced. This effect works to reduce the height of obstacles that can be cleared by a scaled-down jump.

The (negative) work done by gravity during the launch is lower for a scaled down jump than for a normal jump. This effect works to increase the height of the scaled-down jump.

 

[voko]

If you want a practical demonstration of that, fleas can jump 100 times their own height.

That only demonstrates that fleas are not humans.

(Unless you first demonstrate that their muscular tissue is the same as in humans, it is organized into muscles in the same fashion, etc)

 

[sophiecentaur]

Also, fleas will probably have a different velocity ratio in their limbs. I think this is another reason that the scaling wouldn't apply. If you had some running blades, suitably tailored (or leg extensions), you might get away with it, though.

 

[A.T.]

A normal human jump involves pulling up the feet and otherwise contorting the body to clear obstacles without necessarily raising the center-of-gravity a corresponding distance. With a smaller body, the ability to do this will be reduced. This effect works to reduce the height of obstacles that can be cleared by a scaled-down jump.

That's true, but a blender is just about 20cm. A normal human can certainly raise his COM that high during flight. So if that flying-COM-rise is preserved in scaling, the mini-human will escape the blender. Given constant gravity and no air resistance, the flying-COM-rise depends only on the lunch speed upwards.

It seems to me that without gravity (e.g. pushing off a space station), the achievable launch speed would indeed be preserved during uniform scaling with a factor S:

F_muscles ~ S²
d_push ~ S
W_muscles ~ S³
So:
ΔKE ~ S³
m ~ S³
v = const

Introducing gravity we get a negative work term
F_grav ~ S³
d_push ~ S
W_grav ~ S⁴
So now:
ΔKE = W_muscles - W_grav

And since W_grav drops faster than W_muscles with S, the mini-human should be actually able to achieve a higher launch speed, and raise his COM more during flight. At least in vacuum.

With air resistance the mini-human will be slown down more during flight, with approx. a ~ S^-1

 

[Lars Krogh-Stea]

According to this article the volume of muscles is closely related to the the torque of the joints.
http://m.ageing.oxfordjournals.org/content/38/5/564.full
Let's say the volume of your quadriceps muscle is 10 litre, the volume of your body is 100 litre and the density is 1kg/litre. Your upper leg bone is 0,5 metres. Your height is 2 metres. At this height you can jump (move your centre of gravity upwards) 1 metre, or half your height. If we say that 10 liters of muscle creates the torque to generate 1Nm of force, the force that acts on your kneejoint to pushyou upwards will then be 2 Nm. 2Nm to lift your body of 100kg one meter. If we scale you to half the height, in stead of 100x100x100 cm, you would be 50x50x50cm. You will then weigh 12.5 kg. The volume of your leg muscle is now 1.25 litre. The force of which acts on your knee joint is now 0.1(Nm per litre)*1.25/0.25(the new length of your upper leg bone)=0,5Nm. Since you need 2/100=0,02Nm per kilo bodyweight to jump half your bodyheight. you will now be able to propell (0.5/0.2=25) 25kgs to the same height ( Since you weigh half of that you will be able to jump twice as high=100cm.) The result is that if you shrink to half the size, the force your muscles create will be one fourth of the original, but your weight will be one eighth. You will then have twice the jumping power and at half size be able to jump the same height as you did at full size. Any opinions?

 

[jbriggs444]

You are replying to a two year old post. But yes, based on energy considerations and assuming that everything scales identically, the smaller you are, the higher you can jump as a fraction of your own height.

 

[sophiecentaur]

Everyone has heard the old chestnut about a flea and an elephant. Life doesn't scale - even amongst humans.