You're both right!
One car travelling at 80mph does indeed have double the K.E. of two travelling towards each other at 40mph, but in order to see why this is irrelevant you need to look closely at the two theoretical accidents.
In the case of two cars of similar mass hitting head on, the primary impact will (theoretically) stop both cars dead at the point of impact. Thus the energy absorbed by each car is that determined by it's mass and a 40mph change in velocity.
In the second instance, ie one car striking another stationary one, what actually happens is that the initial impact will cause the stationary car to accelerate by half of the original impact speed. So one car slows from 80 to 40, the other accelerates from 0-40. So once again the energy absorbed by each car is that determined by it's mass and a 40mph change in velocity.
There is the added complication that in the second instance the entire mess is now travelling at 40mph, so a secondary collision and further damage is a distinct possibility. Indeed, this is where the other half of the missing energy has gone - there is now the same amount of energy still there as was dissipated in the original collision.
So whether a 80/0 head on is worse than a 40/40 head-on depends very much on what is the other side of the stationary vehicle. If it's an empty road then there isn't much difference. If there are more parked cars there then the eventual effects could double - or even worse if you consider that in the secondary collision your crumple zones might already be crumpled, so any impact is transmitted much more directly to the passengers.
This, incidentally, is a great justification for NOT stopping two feet behind the car in front when all the traffic stops on the motorway. The size of the gap you leave in front can halve or double the effects of someone running into the back of you - yet another illustration of how there is nearly always something you can do to prevent or mitigate an accident, no matter how free of "blame" you are.
Are we all in agreement now?
