Every year I attempt to visit my old PhD supervisor Brian McMaster (old in the sense that my PhD is now a thing of the past, I am making no reference to the man in question!) at Christmas time to have a quick natter and exchange gifts. I was squeezed for time this year since I also had to hire a gown from Queen’s to attend an event for the University of Ulster (long and boring story – getting an award for our work on OPUS). Anyway, just before I left, I asked him about my previous Crashing Cars problem. He wasn’t by any means the first PhD I’d asked about this, and I’d even asked a few physicists. I was hoping of course, that he would immediately say I was being stupid and had missed something obvious, but he found the problem as bothersome as I did.
I continued to mull it over a bit, and even found the problem had some more unsettling properties to do with the masses of the car, but didn’t make any progress unraveling the mystery. Well, a few days later Brian CC’d me on an email to someone with whom he had clearly been discussing the problem with a possible solution. Having read it a good few times and thought it over it makes sense to me and it doesn’t come as a surprise that Brian was the one who cracked the central nub of the problem. I’m a lot more cheerful about it now, but it goes to show that dark nasties can lurk in surprisingly simple problems.
I was brooding some more about the frames of reference thing and maybe beginning to see where the paradox lies. Thing is [perhaps] that the observer in the car is non-accelerated only up to the moment of impact: we can’t use him to assess KE after that moment with the same cavalier abandon that prevailed beforehand. [Especially since he’ll have a headache.] Thus it is not legitimate to say: “from car 1’s POV, KE before = 2mv2, KE after = 0 + 0, therefore KE dissipated into crunch = 2mv2”. Which would be very troubling since it would appear to let us distinguish between states of rest/uniform motion by Physics.
What we can say instead is that from the point of view of a non-accelerated observer *travelling initially with car 1*, the KE before = 2mv2 and the KE after = 1/2 2m (-v)2 = mv2 so the KE going into the impact process = 2mv2 – mv2 = mv2. Which agrees with the observer on the roadside! So maybe the old geezer with the mop of white hair and the century’s most iconic formula was right after all. I’m sufficiently encouraged to copy this email to my tormentor in Jordanstown and see if it allays his apprehensions. Hi Colin!