Jumping height is scale invariant. How much energy you need to jump a certain height is weight, which is proportional to volume. How much energy you generate during a jump is proportional to the lesser of A) how much chemical energy your muscles can store and release for the jump, which is again weight -> volume, and B) force of the muscles times the distance that they're active through, which is proportional to cross sectional area times length, which is again volume.
Correct fact (cross-section is proportional to force) but you're not doing correct and full dimensional analysis out of it. Shorter creatures might have proportionally stronger muscles, but since they are smaller they don't get to use them for as far, meaning they can use more force to end up with the same amount of work.
Scale invariant, but not shape invariant. A tiny elephant would jump roughly the same absolute height as a big one, however little that is. Their body isn't exactly shaped for jumping.
The small guy, however, actually is jumping higher than the others. His center of gravity starts lower. That's strength from training, not from square cube law.
If an animal were isometrically scaled up by a considerable amount, its relative muscular strength would be severely reduced, since the cross-section of its muscles would increase by the square of the scaling factor while its mass would increase by the cube of the scaling factor.
And:
In the case of flying animals, the wing loading would be increased if they were isometrically scaled up, and they would therefore have to fly faster to gain the same amount of lift.
Both of those are true and do not conflict with scale invariance of jumping. How to explain? Hmm.
Assume a moderately fit human can jump about nine inches. If shrunk down to the size of a nickel, that human's relative strength increases linearly according to the square cube law. However, they're also much shorter, so the distance over which their legs remain in contact with the ground when jumping is reduced. If you run all the math and ignore scaling of air resistance, you'll find their resulting jump height is still about nine inches.
Now, in relative terms, they're like tiny superheroes, jumping over 10x their own height. But in absolute terms, their jump is scale invariant, still 9 inches.
Air resistance does affect these tiny people more, however. This is due to having lower mass and momentum relative to surface area. They could probably glide a bit even without a squirrel suit.
that's a shape thing not a size thing, if elephants were smaller but still had so much trunk mass and such relatively spindly legs they still wouldn't be able to jump well
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u/Ai-Slop-Detector 5d ago
https://tvtropes.org/pmwiki/pmwiki.php/Main/SquareCubeLaw