Rail Car with a Sail in the Wind

[MaratZakirov]

Hello everyone, I must admit that I recently failed to find exact solution for equations I wrote above. I can of course make a brute-force or even gradient based solution, but exact analytical solution I can not, I am already spent too much time for it. So if you have some ready to show solution, I'll be very thankful for that.

 

[haruspex]

Hello everyone, I must admit that I recently failed to find exact solution for equations I wrote above. I can of course make a brute-force or even gradient based solution, but exact analytical solution I can not, I am already spent too much time for it. So if you have some ready to show solution, I'll be very thankful for that.

I find it easier to drop the vector form for a problem like this. I set the angle of the sail to rail as theta, incoming particle to sail as phi, departing angle to sail as psi, incoming speed u, departing speed u', cart's acquired speed v.
I got a rather messy quadratic in x where $x=\frac{u'^2}{u^2}$.

Please post your working as far as you get.