@Ronald Raygun
Suppose car speed and wind speed are *both* 55ft/s. The
wheels are still delivering 1hp. The headwind is zero, so you
would calculate the power needed by the prop to produce 10lb
of thrust as 0hp. Can't be right.

But it is right, because at that point in the calculations we are
still describing the scenario using theoretical, lossless components.
We've defined the transmission method as lossless and the propeller
also as 100% efficient.

At 0hp the prop won't be turning, and would be producing no thrust.

If the vehicle is traveling at exactly windspeed, the propeller is
working in what's called 'static conditions' and according to standard
propeller efficiency terms is doing no work (different "efficiency
terms" are used for fans).

In static conditions, a 100% efficient propeller would be infinitely
large and would be turning infinitely slow but would still be
producing 10lbs of thrust at 0hp. Its not unlike leaning a 2x4
against the wall – it can happily exert 10lbs of force on the wall all
day long while needing 0hp to do it. Same with a lossless propeller
in static conditions.