A car drives along a long straight road, with its engine producing constant power $P_0$. A headwind blows against the car with instantaneous speed $v_{\rm wind}$, fluctuating about a mean value $v$ (with $v_{\rm wind}$ always positive). The air drag force on the car is $Av_{\rm rel}^2$, where $A$ is a constant and $v_{\rm rel}$ is the relative speed between the car and the air. If the wind were perfectly steady at $v_{\rm wind}=v$, the car would settle to a steady speed $v_0$ satisfying $P_0 = Av_0(v_0+v)^2$. The car's speed adjusts quasi-statically to the wind fluctuations (it is always at its instantaneous equilibrium speed). What is the time-averaged car speed $\overline{v}_{\rm car}$?