Again, not trying to calculate the power in the wind. We are calculating the power to move the object in the wind. That power is the force times the vehicle velocity. You can dance around all you want, but the fact remains you are applying the wrong equation to this.
Imagine the sail is stationary and the wind blows on it. The force on the sail is applied through linkages to the vehicle. Now the vehicle is not in the wind, only the sail. Clearly in this situation the force that must be applied to the vehicle to maintain a constant velocity is the force on the sail from the wind and the ONLY velocity that is relevant to the vehicle is the vehicle speed, so
0.5 * air density * area * (wind speed - vehicle speed)² * (vehicle speed)
I've explained it, I've derived it. You are not capable of understanding it or you are just refusing to acknowledge you understand it in public.
We are trying to calculate the available wind power to the vehicle as that is all the vehicle can use since it is a wind only powered vehicle thus the most important equation to get right.
The vehicle will not be stationary if there is no friction including no brake. The lowest energy state for that vehicle will be vehicle speed = wind speed.
If vehicle without brakes wants to maintain zero speed relative to ground it will need to provide this power
0.5 * air density * area * (wind speed - vehicle speed)² * (wind speed - vehicle speed)
And since you want vehicle speed to be zero relative to ground you have 0.5 * air density * area * (wind speed)
3This shows that even an ideal vehicle able to get all wind power can at most just use that to stay in the same place as if it will want to move upwind it will need that plus something extra.
As for direct down wind the ideal vehicle can get to same speed as wind speed but not above that without having some stored energy or an external energy source other than wind.
You all seems to agree that is correct for a sail vehicle but refuse to believe that is valid for any other type of vehicle including blackbird.
Blackbird is not using wind power when above wind speed but it is using earlier stored wind energy and the difference is stored energy is a limited resource so a real vehicle with friction at some point will start to slow down.