Why? No production car is brake limited; slam on the brakes in any car and you'll go into (and stay in) ABS, meaning you're at the limit of tire grip, and therefore at roughly constant deceleration. Where would energy dissipation come into the picture? The energy is being dissipated by the brakes, not the tires, and they're not the limiting factor.

There is a different, clever way to analyze the problem thinking about energy and work.

Work is by definition (work) = (force) x (distance), or the same thing expressed as an integral.

Distance is the same, that’s a parameter of the problem. If we say that the brakes apply constant deceleration, then that implies that the force is the same, between both cars. Then the amount of energy shed by both cars is the same before they reach the barrier. (We don’t need to use the integral definition for work here, because force is a constant.)

This explains why the people who assumed that both cars dump the same amount of energy arrived at the same result, even though some of the people in this thread appear to be using faulty reasoning.

Ah, of course, you're right. Thanks for pointing that out! I didn't really think about it at the time, but the equation you end up with if you use the acceleration approach results in a v^2, which makes sense when you consider that the equal energy dissipation approach is equivalent.

But the scenario we're talking about here isn't repeated hard braking.

brakes heat up, and the hotter they get, the faster they dissipate heat. nobody here has taken that non-linearity into account.

The relationship between temperature and friction varies for different brake pads. Of course everything dissipates heat faster the hotter it is, but only racing pads actually tend to gain friction with heat (up to a point).

But again, it doesn't matter, because road cars are traction limited in braking.