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...