Why can bugs sustain falls from very large heights (relative to their size)? Is it because their mass is so small that the force with which they hit the ground isn’t that much relative to the amount of force their bodies can sustain?read more
My friend's answer.
That's part of it, but not the whole story. If you write out the force equation for the bug, you get
gravity pulling down + drag pointing up = mass * acceleration
mg - drag = ma
bugs are small, so what happens with this equation when m is small? Can't tell, since we don't know about drag. So let's instead vary size of bug, not mass. the bug has mass density p and diameter d (call it a cube, who cares), so
m = p*d^3
so now we have
p*d^3*g - drag = p*d^3*a
drag we know must involve the surface area and velocity, so we'll say that
drag = c*v*d^2, for some unknown drag coefficient c
finally, plugging in drag:
p*d^3*g - c*v*d^2 = p*d^3*a
p*g - v*c/d = p*a
drag increases with velocity, so there must be a terminal velocy w such that drag exactly balances gravity, and so a = 0:
p*g - w*c/d = 0
w = p*g*d/c
very cool! this means that the object's terminal velocity increases with density (styrofoam cube vs iron cube), gravity, and diameter. Very roughly speaking, we can say that a bug is mostly water and we are mostly water, so say that a bug's density is the same as a human's. I would bet that our drag coefficients are also similar. So for bug(1) and human(2):
w1 = p1*g1*d1/c1 = p * g * d1 / c
w2 = p2*g2*d2/c2 = p * g * d2 / c
So the only thing that makes a difference is the diameter! And of course a bug's d is smaller than a human d, so the bug will have smaller terminal velocity.
With sharp impacts of two solids, usually the way to go is to say that the impulse = integral F(t) dt = change in momentum, where F(t) is the force on the bug during impact -- this normally looks like a gaussian. So, roughly speaking again, a bug will have smaller gaussian because its terminal velocity is smaller. A smaller gaussian means smaller maximum acceleration on the bug.
That's the first half. The second half is what you were getting at. Ants can carry ~50x their body weight, which means that their body can have 50x body weight put on it with no problem. This is obviously not true of humans -- maybe a few times bodyweight at most. So it's safe to say that the insect body can also withstand a higher relative force than humans can.
So, for some impact with the ground, the maximum force endured during impact is more likely to be less than the maximum allowable by the insect.
Sorry for the wordiness.