A car of mass m and velocity v has an energy of
E(1) = 1/2 mv^2.

So the combined energy of two identical cars at identical velocities is simply twice as much as it is for one car, or E(2) = mv^2. It doesn't matter which direction they're travelling, relative to each other or to a common coordinate system. The velocity of each is still v.

Kinetic energy is a linear function of mass, but a function of the SQUARE of the velocity, if you add up the velocities you would get E(3) = (1/2)(2m)(v+v)^2 or E(3) = mv^4. But the latter solution introduces a paradox; why should two bodies at a given speed and velocity always have a given total energy, unless they are aimed at each other? Why should the energy of both cars suddenly double just because they are colliding? Where does the extra energy come from?

The answer, of course, is that there is no extra energy. In each case, the energy is calculated not with reference to each body, but to a common coordinate system (in this case, the coordinate system in which the point of impact is stationary.) You add up the energies for each car, you don't add up the velocities.

It doesn't take an Einstein, a Newton will do.

I hope they solved it experimentally. Head-on collisions are more fun to watch.

Moral: common sense isn't always right. The world is more complicated than you are.

Now, a related problem. A lightsail spacecraft is propelled by light photons striking the sail and imparting thrust to it. To maximize this impulse, should the sail be painted black, so that photons are absorbed, or should it be painted a highly reflective color so the photons are reflected?