Path for minimal time of river crossing (Top Contributors only I guess)?

Question

Here's a problem I solved this weekend but for which I would like to see possible other solution methods.

You have a river of width W, where the water flow has parabolic velocity profile:

v(x) = 4 v_middle (x/W) ( 1 - x/W)

[[The river flows in the +y direction]]

Your boat has a constant speed C relative to the water, where C > v_middle .

Determine the path y(x), where (0, y(0) ) is the departure point, that minimizes the time for crossing the river

(i) If you have to arrive at a point directly opposite to your point of departure
(ii) If you do not need to end directly oposite, but just reach the other bank.

I solved it using Variational Calculus, but am curious about other possible solution methods.

So, top contributors, consider this to be your challenge of the day...

supastremph · Accepted Answer

Variational calculus is a natural method for solving this problem, it's what I would use, so I can't help you formally.  

But I did notice number two can be solved even more naturally--without math at all.  The y and x axes are orthogonal, so if you don't care where you end up on the other side (or perhaps more importantly for i) if C were < v_middle) you just always stay pointed at the other side.  This is the maximum of C dot xhat.  

I'm interested because I haven't done one of these in a while.  How did you set up the functional, did you employ the constraint of y(W)=0 via a Lagrange multiplier?

lemoi · Answer

You and another contact have both lost TC badge now. i have not checked some thing else, yet this makes no experience. You both were asking and answering. only checked and four contacts are lacking their TC badges.

Path for minimal time of river crossing (Top Contributors only I guess)?

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