A man wishes to cross a river of width 120 m by a motorboat. His rowing speed in still water (or relative to water) is 3 m/s and his maximum walking speed is 1 m/s. The river flows with a velocity of 4 m/s.
(a) Find the path which he should take to reach the point directly opposite to his starting point in the shortest time.
(b) Find the time required to reach the destination.
2. Homework Equations
Let ω be the width of the river and x be the drifting of the boat.
Let v denote the velocity vectors and v denote their respective magnitudes.
vr = absolute velocity of river
vbr = velocity of boatman relative to the river or velocity of boatman in still water
vb = absolute velocity of boatman
Hence, vbr = vb - vr
∴ vb = vbr + vr
Time taken to cross the river t = ω/vbr-y = ω/vbrcos θ
Drift x = vb-xt = (vr - vbrsin θ)ω/vbrcos θ
The Attempt at a Solution
Now, in order to reach the point directly opposite to his starting point, resultant velocity vb must be perpendicular to the river current.
So, for this the drift x must be 0.
x = 0 gives vr = vbrsin θ
Subsituting, sin θ = 4/3, which is meaningless.
I can't proceed without θ. Also, I don't get why they've given his maximum walking speed.
Am I missing anything?
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