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Question

A boat moves relative to water with a velocity which is n=2.0 times less than the river flow velocity. The angle (in degrees) to the stream direction must the boat move to minimize drifting is (120+x) . The value of x is :

Solution
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Let v0 be the stream velocity and v the velocity of boat with respect to water. At
v0v=η=2>0, some drifting of boat is inevitable.
Let v make an angle θ with flow direction (shown in figure below), then the time taken to cross the river
t=dvsinθ (where d is the width of the river)
In this time interval, the drifting of the boat
x=(vcosθ+v0)t
=(vcosθ+v0)dvsinθ=(cotθ+ηcscθ)d
For xmin (minimum drifting)
ddθ(cotθ+ηcscθ)=0, which yields
cosθ=1η=12
Hence, θ=120
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