# How to form equations of motion of this kind?

I needed help with this question. I don't know how to form the equation of motion that describes the path of the boat in the following problem.

A boat which is rowed with constant velocity $u$ starts from point $A$ on the bank of river which flows with a constant velocity $v$ and it points always towards a point on the other side exactly opposite to $A$. Find the equation of path of boat.

In my paper they gave the final answer but did not show the steps on how it was derived. The answer which they gave was $r \sin(θ) = c\left(\tan\left(\frac{θ}{2}\right)\right)^{u/v}$ where c is a constant.

Also i have no idea what $r$ is in the answer although i think it might be the distance between the points $A$ and $B$ but $U$ could be wrong. I think $θ$ might be the angle by which the boat deflects.

• You need to include a simple diagram, as well as setting out clearly why you think what you say above is correct – user108787 Oct 24 '16 at 0:18
• This sounds like a variational calculus problem. – John Alexiou Oct 24 '16 at 0:32
• Possible duplicate of Trajectory of a swimmer trying to reach the opposite bank. – sammy gerbil Oct 24 '16 at 2:43
• It looks like that question but the answers are different. Is it because of different integration meathod ? – Kashish Oct 24 '16 at 3:16

Given that your question is partial, I decided to create a polar plot of the function and help you understand the equation.

(For this explanation assume that the river is 5 meters wide.)

This plot shows the curve with the following parameters: c=0.1, u=9, v=1.

So, to interpret the equation of motion in this case, the boat moves as one kinda expects: it heads toward the opposite bank and is carried by the river.

The key to understanding how this equation works is internalizing that the boat does not point straight at the other side at all times: as it moves down the river due to the river current, the boat constantly changes direction. Thus, less and less of the boat's velocity is used to travel to the other side of the bank.

This next plot shows the curve with the following parameters: c=0.1, u=1, v=1.

Notice that by making the velocities equal the boat appears to never reach the other side of the river.

Edit: I looked at Sammy's link and here is the line where I would work from:

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Move -v/u to the other side and raise e by both sides thus removing the natural log (you'll end up with a new constant c).

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Use the following identities:

and

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With those identities you'll be able to get the equation; and don't forget that .