So, over at an astronomy forum, a user called BetaDust asked why in the famous E=mc2 equation, the speed of light is squared?
One of the answers given was a reasoning-by-analogy sort of response, by someone with username DaveC followed by his inmate number, who cites the formula for kinetic energy, containing velocity squared. He further explains that this is why a 40-km/h collision is much more devastating that a 20-km/h collision, because the amount of energy involved is four times as much.
It is this response which I am thinking about now, and something about it bothers me. I am possibly coming to the conclusion that it has the essence of the truth, but there are some hidden assumptions here. So I want to do some thinking out loud, and see what everyone says.
So, the basic argument is, kinetic energy is m v2/2, so if you double the velocity of the car, you quadruple its kinetic energy. So the transfer of kinetic energy is four times as much. This sounds reasonable, and it is a type of argument I have thought out (and possibly stated) myself before. Except . . .
Suppose the car is passing another car driving at 20km/h, and then crashes into the brick wall in the passing lane. Since there is brick wall in the non-passing lane, the car that was passed continues unimpeded.
Then from the perspective of an observer on the side of the road, the car that crashed went from 40km/h to 0km/h. So the kinetic energy changes from 800*m to zero.
However, from the perspective of the driver of the car that was overtaken, the crashing car was travelling at +20km/h, and then -20km/h after the collision. So the kinetic energy changes from 200*m to 200*m. Even if we allow for a directional component, we can view the kinetic energy as changing from 200*m to zero, and then 0 to 200*m, for a total change of 400*m. So half as much transfer from this perspective. Obviously the amount of damage cannot depend on the perspective of the observer, so something is missing here.
To eliminate the directional issue, let us make it a spacecraft flying through open space at 40km/h, and crashing through a wooden barrier that slows it down to 20km/h. Then the change in kinetic energy is from 800*m to 200*m, so a net change of 600*m. However, from the perspective of a spacecraft travelling 20km/h that had just been overtaken, the crashing spacecraft changes its velocity from 20km/h to 0. So the kinetic energy changes from 200*m to 0. So the change in kinetic energy is three times larger or smaller, depending on perspective. The damage done by the collision must be invariant, though. So again, something is missing from the analysis.
So the only two possibilities I can come up with here are, the idea that the damage done by the impact will be proportional to the kinetic energy transferred, is just wrong, or we need to take into account the change of kinetic energy in the object that is stricken as well.
So, it's a bit tedious, but when I take into account the change in kinetic energy in both the car and the barrier, I get that this is invariant. Note that the change in the velocity of the barrier can be calculated using conservation of linear momentum.
So the conclusion I am going to is that DaveCXXXXX's explanation may be the right idea, but it is somewhat incomplete (or perhaps just rests on an unstated assumption). Then the barrier changes in velocity from zero to <very small>, the change in kinetic energy is <very small> squared, which is <super small>. But when the barrier changes from <a lot> to <a lot>+<very small>, when we square the velocity, we get a term proportional to <a lot>*<very small>, which is <not negligible>. So when we change perspective, the amount of kinetic energy lost by the car/spacecraft changes, but the different is picked up by the barrier, so the total change is invariant.
So does that sound reasonable? The total change in kinetic energy is invariant of perspective, and this loss of energy is what goes into doing the damage. How the damage is divided up between the car/spacecraft and the barrier, this I do not know.