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.

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

2 Answers