It seems that more mass is expelled at equal or higher acceleration if an apparent wind is blowing in same direction as the prop's expelled air mass than when an apparent wind is blowing in the opposite direction. Therefore the prop thrust is helped by the"same direction" head wind given that the force or thrust is proportional to the mass and its acceleration (f=ma).
The problem here is that the acceleration of the air will be less. The faster the air already moves backwards, the more difficult(more energy required) it becomes to accelerate it even more.
The relevant formula here is: power = force * velocity
With more velocity, you need more power for the same force.
Even your equation seems to support this. Holding all else constant,
This is tricky. The prop_efficiency is not constant but highly dependent on the conditions. It can achieve 80-90% at design conditions. But for most of the run it is far worse.
the equation you gave says that thrust is inversely proportional to apparent air speed meaning more wind speed leads to less prop thrust if the vectors of the wind and prop thrust are aligned. It follows that if the apparent wind vector is opposite the prop thrust then it subtracts from the prop thrust so less of the apparent tail wind is better for the prop thrust.
This is even more tricky. The prop formula (coming from the realm of aircraft propellers) offers a sensible definition of propeller efficiency in apparent headwind (the normal case for planes). To use it for airspeeds <= 0 you would have to assume negative efficiency, because the propeller is now slowing down the air (in the propeller's frame). The formula doesn't really help us to calculate thrust in the below windspeed situation, because it assumes a certain direction of energy flow, which now reversed. And there is little empirical data on propellers in this condition.
This would explain why it has such a slow start and then accelerates to faster and faster speeds
This is the variable propeller efficiency. Initially the blades are stalled (separated flow). When it comes closer to the design conditions, they unstall and work more efficiently.
So I'm still uncomfortable hearing that "The apparent head wind never helps to accelerate. "
In general, it would make no sense, if the relative headwind, that you create with your own motion, would help you to accelerate.
In general, in terms of optimally achievable performance:
relative headwind : bad
less relative headwind : good
But for a specific vehicle, for example with fixed propeller pitch, there is one optimal combination of airspeed and ground speed, that offers the best attack angle at the blades. So sometimes an increase in airspeed can increase the acceleration. But not by increasing thurst, but rather by increasing the prop efficiency and thus decreasing the wheel drag:
better attack angle at the blades ->
less induced drag at the blades ->
less reaction torque that brakes the wheels
But that seems inconsistent with F=ma if more head wind means more mass expelled so more thrust and not less.
See first paragraph.