In SpaceX's webcast today they talked about the first stage getting back to Earth with twice the velocity as usual. I quote: "So 4 times as much energy means harder on the stage, and 8 times as much heating." I understand 2x the velocity means 4x as much energy, E=1/2mv^2, but why does that produce twice as much heat relative to the energy?
Air friction is roughly proportional to the (magnitude of the relative) velocity squared. Power is equal to force times velocity (the time derivative of work, which is force times distance). Since the force is proportional to the velocity squared, then the power will be proportional to the velocity cubed.