I am a mathematician and know nothing about physics, although I am trying to solve a problem related to physics. If someone could help or provide some suggestions, I would be grateful.
I have a speedy boat navigating in the sea. The boat could be very small and not so heavy. Some breeze blows, and there exists a current. I assume that the resultant vector force affecting the boat's navigation is $W$. This boat starts its navigation from a special point, let it be $p$, and can choose every direction to move with a constant velocity related to the water. I also assume that the conditions are uniform across the sea, meaning the wind, the current, the temperature, water depth, etc., are equal at all points on the surface of the water.
What would be the locus of the endpoints of the velocity vector of the boat after one unit of time?
It is easy to see that if the water is at rest and no wind blows, this locus is a circle. Now, my question is, in the conditions of this problem, could the locus be an ellipse with its major axis in the direction of W?
I tried to approach the problem using Zermelo's problem of navigation. The problem is that we need some metric on the space to find the equation of the locus. What I want to do here is to find the metric using the locus.
P.s.: By a unit of time, I mean one second, one minute, or one hour.
Thanks in advance for every comment, suggestion, or answer.