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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.

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We know from Newton's second law that

$\displaystyle \vec F = \frac {d(m \vec v)}{dt}$

and as long as the mass $m$ of the boat is constant we can conclude that

$\displaystyle \vec F = m \frac {d \vec v}{dt}$

which we can re-arrange to get

$\displaystyle \frac {d \vec v}{dt} = \frac {\vec F} {m}$

If the force vector $\vec F$ is constant (in both magnitude and direction) then we have

$\displaystyle \vec v(t) = \vec v(0) + \frac t m \vec F$

So if the locus of $\vec v(0)$ is a circle then the locus of $\vec v(t)$ will be the same circle displaced by $\frac t m \vec F$.

If $\vec F$ depends on $t$ but not on $\vec v$ then we have to find the integral $\vec G(t) = \int_0^t \vec F(t) dt$, and then

$\displaystyle \vec v(t) = \vec v(0) + \frac 1 m \vec G(t)$

once again, if the locus of $\vec v(0)$ is a circle then the locus of $\vec v(t)$ will also be a circle. If we want the locus of $\vec v(t)$ to be a different shape from the locus of $\vec v(0)$ then we need a force $\vec F$ that depends on $\vec v$ - such as a drag force.

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  • $\begingroup$ @ganlalf61 thank you so much! $\endgroup$
    – Majid
    Commented Apr 1 at 1:00
  • $\begingroup$ gandalf61, what happens if the boat is so light? Or as another problem, what happens if instead of a boat we have the leaf of a tree? That is we have a leaf on the surface of a pond and the breeze blows we want to study the velocity vector of the leaf motion. I mean how to find v(t)? We should consider what for m? $\endgroup$
    – Majid
    Commented Apr 1 at 11:55
  • $\begingroup$ In fact, in the case of the leaf on the water surface, the leaf itself does not have any velocity vector and its motion depends on the water flow and the wind. I am wondering if it is the same problem as the speedy boat. $\endgroup$
    – Majid
    Commented Apr 1 at 11:59
  • $\begingroup$ and one more comment! You mentioned that if the locus of v(0) is a circle! Under the assumption that the temperature, depth of water, and all the conditions are the same in all the points, the locus might be something else? $\endgroup$
    – Majid
    Commented Apr 1 at 12:57
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Perhaps this result can help you ?

for a 2D simple dynamic boat simulation where the velocity is constant and you have side wind disturbance , the y via x position of the boat look like this

enter image description here

the boat yaw angle

enter image description here

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  • $\begingroup$ the spinal curve from inside is the locus of endpoints of velocity vectors? $\endgroup$
    – Majid
    Commented Apr 4 at 0:12
  • $\begingroup$ This is the path of the boat , die velocity is constant,It look like Logarithmic spiral $\endgroup$
    – Eli
    Commented Apr 4 at 6:15
  • $\begingroup$ Sorry, I do not understand it. The wind is some kind of rotational wind? $\endgroup$
    – Majid
    Commented Apr 4 at 16:43
  • $\begingroup$ Ah I see. Yes of course. Interesting! $\endgroup$
    – Majid
    Commented Apr 4 at 21:28

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