I saw this on MythBusters. :smile:
Assuming a totally inelastic collision, there is equal energy transfer between both cars. Similarly because of the disparity in mass of the mountain and the car, whatever pitiful amount of kinetic energy is transferred by the car into the mountain, the...
I agree that the equation does seem to appear out of nowhere, so it's probably just a given that the equation x=0.08t^3 describes the car's motion. The author probably should have pointed this out. It doesn't really follow that this is the case from any of the text that precedes it.
HTH