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I already asked a question very similar to this one here and I think the solution would not work when the boat navigates in the sea when a very strong wind blows. That is I am trying to find the answer to the following:

I have a speedy boat navigating in the sea with the constant velocity concerning the water (maximum power of its engine). A strong wind blows and the total force of W affects the boat. I assume that all the other conditions, depth of water, temperature, ...., are identical in all directions and points of the sea and the wind magnitude and direction do not change. The boat can move in every direction.

What would be the locus of the endpoints of the velocity vector of the boat after one second when it starts its navigation from a point?

I am wondering if Newton's second law can solve the problem.

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  • $\begingroup$ Since you stipulate that the velocity of the boat in the water's frame ($\vec v_b$) is constant, nothing else matters. The velocity is $\vec v =\vec v_b + \vec v_{current}$ in all frames, assuming $v_i \ll c$, so the answer is $\vec P + \vec vt$. $\endgroup$
    – JEB
    Commented Apr 4 at 18:11
  • $\begingroup$ @JEB what are v_i and c? $\endgroup$
    – Majid
    Commented Apr 4 at 18:32
  • $\begingroup$ the former is an abstract subscript standing for all velocity subscripts, and $c=1$ is, as aways, the speed of light. $\endgroup$
    – JEB
    Commented Apr 6 at 15:05

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enter image description here

if the velocity $~v~$ is constant the boat has two generalized coordinates, the side slip angle $~\beta~$ and the yaw angle $~\psi~$.

the equations of motion are

$$\dot\beta=-{\frac {\sin \left( \psi+\beta \right) {\it F_x}-\cos \left( \psi+ \beta \right) {\it F_y}}{mv}}$$

$$\ddot\psi=\frac{\tau_\psi}{I_\psi}\quad, \tau_\psi=L\,(F_x\,\sin(\psi)-F_y\,\cos(\psi))$$

where $~F_x~,F_y~$ are the components of the wind force given in inertial system.

to obtain the position of the center of mass in inertial system you need additional two differential equations

$$ \dot x=v\,\cos(\psi+\beta)\\ \dot y=v\,sin(\psi+\beta)$$

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  • $\begingroup$ and once one solves these two equations, (x,y) in the position vector? $\endgroup$
    – Majid
    Commented Apr 5 at 23:40
  • $\begingroup$ You don’t have analytical solution $\endgroup$
    – Eli
    Commented Apr 6 at 7:52

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