So, I think the objective is to maximize the component of velocity ##v_x## of the cart by maximizing the force ##F_x##. A maximal ##v_x## maximizes ##v_y## by constraint (3) in post
#103.
To achieve this, I introduce the parameter ##\varphi## for the angle of rotation of the vane relative to the cart. The intention is to find ##\varphi## as a function of ##v_x## that maximizes ##F_x## for a fixed angle of turn ##\beta##.
From the diagram above we find that:
$$ F_x = \dot m \left( v_x - w \sin( \beta + \varphi )\right) \tag{1}$$
Next, attempt to optimize ##F_x## w.r.t. ##v_x##
$$ \frac{dF_x}{dv_x} = \frac{ d \dot m }{d v_x} \left( v_x - w \sin( \beta + \varphi ) \right) + \dot m \left( 1 - w \cos ( \beta + \varphi ) \frac{ d \varphi}{d v_x} \right) \tag{2}$$
Under continuity the mass flowrate entering-exiting the control volume is given by:
$$ \dot m = \rho A ( w + v_y ) = \rho A ( w + v_x \tan \theta ) \tag{3}$$
This implies that:
$$ \frac{ d \dot m }{d v_x} = \rho A \tan \theta = constant. \tag{4} $$
Making the substitutions ##(3)## and ##(4) \to (2)## becomes:
$$ \frac{dF_x}{dv_x} = \rho A \tan \theta \left( v_x - w \sin( \beta + \varphi ) \right) + \rho A ( w + v_x \tan \theta ) \left( 1 - w \cos ( \beta + \varphi ) \frac{ d \varphi}{d v_x} \right) \tag{2'}$$
Optimization implies looking for the solution to:
$$ 0 = \frac{dF_x}{dv_x} = \rho A \tan \theta \left( v_x - w \sin( \beta + \varphi ) \right) + \rho A ( w + v_x \tan \theta ) \left( 1 - w \cos ( \beta + \varphi ) \frac{ d \varphi}{d v_x} \right) \tag{5}$$
Solving (5) for ## \varphi '## yields the following first order nonlinear ODE:
$$ \frac{ d \varphi }{ d v_x} = \frac{w - w \tan \theta \sin ( \beta + \varphi)+ 2 \tan \theta v_x}{ \cos ( \beta + \varphi ) \left( w^2 +w v_x \tan \theta \right) } \tag{6}$$
How am I doing so far?