> Keep in mind that when falling your acceleration is 9.8 m/sec/sec, so a 10 inch fall is worse than 10 times as bad as one inch.
Shouldn't that be sqrt(10) times as bad?
The distance fallen under constant acceleration g for t seconds is 1/2 g t^2, so it takes sqrt(10) times as long to fall 10 times as far. The velocity after falling t seconds is g t, so falling sqrt(10) times as long gives a velocity of sqrt(10) times as much.
It also depends on whether 'badness' is a property of energy or velocity. Velocity goes with sqrt(distance) as you've demonstrated, but energy is directly proportional to distance in this case.
You're right about the velocity, but the impact force really depends on the properties of the material one lands on.
First consider that our kinetic energy is
K = mv^2/2
Additionally is the material has a spring constant k, then the force that results when the material deforms by x amount is
F(x) = kx
If we let d be the max distance the material, then we can integrate over the force to get the energy absorbed, which is equal to the kinetic energy.
int_0^d F(x) dx = K
kd^2/2 = mv^2/2
Now there are two ways the impact could work. Imagine we have a really thick crash pad, then we consider it to deform without limit and have a constant k, giving
d = sqrt(mv^2/k)
F_max = F(d) = v sqrt(mk)
But what about a helmet? It can only over it's thickness, at which point your head is basically in contact with the concrete. Thus, we have a constant deformation distance d, and get
k = mv^2/d^2
F(d) = mv^2/d
As we can see, depending on how we consider it we get either a force of sqrt(10) or 10 times the original amount. I probably failed to take into account something that someone who knew more about materials could point out.
tl;dr it's complicated because impact force does not necessarily scale linearly with velocity.
update: so after some conversation with others. It seems that head on concrete is better modeled by a constant d. So a drop of 10 times the height results in 10 times the force.
Well, it's still worse than 10 times as bad because there are elasticity effects. You can see this on your nose. There is a force you can use to squeeze it that won't do anything but much more and you can break the blood vessels in it and give you a bruise. You could do that first thing many times but there's a threshold effect.
Or a balloon. It recovers from lots of pressing, even repeatedly. But there's a pressure from which it's never going to recover and crossing that threshold has binary outcome.
Shouldn't that be sqrt(10) times as bad?
The distance fallen under constant acceleration g for t seconds is 1/2 g t^2, so it takes sqrt(10) times as long to fall 10 times as far. The velocity after falling t seconds is g t, so falling sqrt(10) times as long gives a velocity of sqrt(10) times as much.