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.

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

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