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.

See also http://hyperphysics.phy-astr.gsu.edu/hbase/impulse.html

The badness might be proportional to the energy being transferred through your head, which is simply proportional to the height you fell before stopping suddenly.

The square root in the velocity calculation goes away since

E_k = 1/2 MV^2

so the energy dispersed hitting the ground from a 10 inch fall is exactly 10x as bad as a 1 inch fall.

Of course the actual damage done by that energy transferred through your head will be some complex function due to bone breakage and cell tearing thresholds, etc. But as a first approximation of suck, I'd go with this simple linear model.

Ha, sorry, yes. I confused 10 times as long with 10 times as far... I should have double checked the math.

Still dropping your head an inch (painful) is a nice reminder of how much energy is involved from normal riding height.

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.