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...

Steve4Physics · Accepted Answer

For neatness I’ll use:
v₀ for v_middle and
A = 4v₀/W
B = 4v₀e/W²
Giving:
vy(x) = Ax - Bx² (equation 1)
where vy is the water’s velocity in the y direction.

I can’t see a clever method to do question (i) but the 2nd question is relatively straightforward I think.

For minimum crossing-time, irrespective of final position, the velocity of the boat relative to the water (C) must always be in the +x direction (else the x component of velocity will be less than C). Then the x-coordinate of the boat is:
x(t) = Ct

In this case, the y-velocity of the boat will be the same as the y-velocity of the water.
Substituting for x in equation 1:
vy(t) = A(Ct) – B(Ct)² 

vy(t) = dy/dt  = ACt - BC²t²

Integrating to find the boat’s y coordinate as a function of time:
y(t) = ½ACt² - BC²t³/3 + K

When t = 0, y = y(0) so:
y(t) = ½ACt² - BC²t³/3 + y(0) (equation 2)

Since x = Ct, t = x/C, so substituting for t in equation 2:
y(x) = ½AC(x/C)² - BC²(x/C)³/3 + y(0) 
= ½(A/C)x² - (B/C)x³/3 + y(0)

Replacing A and B:
y(x) = (2v₀/(WC))x² - (4v₀/(3W²C))x³ + y(0)

I hope that's the correct answer!

lishego · Answer

You and another touch have the two lost TC badge now. i haven't checked the rest, yet it quite is mindless. You the two have been 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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