Work done on a moving cart in wind

Let us suppose a cart is moving in a rough ground with a propeller attached to the roof of the cart. Let the velocity of the cart at the given point of analysis be V and velocity of wind is -W. The force applied respectively by the ground on the wheel and the wind on the propeller be Fwheel and Fprop.

If we want to conserve power for the cart in the frame of reference of the cart (since kinetic energy of the cart in this frame is always 0), I am trying to understand why the following analysis is valid ?

$$P_{input}=F_{wheel}V$$

$$P_{output}=F_{prop}(V-W)$$ [relative motion between propeller and wind]

$$P_{input}=P_{output}$$

$$\Rightarrow F_{prop}=\frac{F_{wheel}V}{V-W}$$

In the above analysis aren't we mixing the velocities between ground frame and the frame of cart? If we consider the points of application of force on the cart (from the frame of the cart),that is the wheel and the propeller, aren't they all fixed to the cart frame and hence no work is being done on either of them?

Isn't $$P_{in}=P_{out}=0$$ and no relationship between $$F_{prop}$$ and $$F_{wheel}$$ can be concluded from power conservation in cart frame?

• youtu.be/yCsgoLc_fzI?t=662 (just for reference) Jul 2 '21 at 13:35
• A reference frame attached to the cart is unlikely to be an inertial reference frame. If you don't include the d'Alembert force(s), your equations are wrong. Jul 2 '21 at 15:18
• Isn’t the Psuedo forces acting on the Centre of Mass, which also have zero displacement and hence zero effect on power? Jul 2 '21 at 15:38