A river of width `w` is flowing such that the stream velocity varies with `y` as `v_(R)=v_(0)[1+(sqrt3-1)/wy]`, where `y` is the perpendicular distance from one bank.A boat starts rowing from the bank with constant velocity `v=2v_(0)` in such a way that it always moves along a straight line perpendicular to the banks.
At what time will he reach the other bank

Options

A. `t=(wpi)/(6v_(0))`
B. `(wpi)/(6(sqrt2-1)v_(0))`
C. `(wpi)/(6(sqrt3-1)v_(0))`
D. `(wpi)/((sqrt3-1)v_(0))`

Answer

C

Explanation

Given `|vecv_(R)|=v_(0)[1+(sqrt3-1)/wy] and |vecv|=2v_(0)`
Resultant velocity of boatman should be along `AB` or perpendicular to `AB` components of `vecv` and `vecv_(R)` should be zero.Hence `v cos alpha=v_(R)`
or `(2v_(0))cos alpha=v_(0)[1-(sqrt3-1)/wy]`
Therefore, resultant velocity along `AB` is `V_(y)=v sin alpha` or
`(dy)/(dt)=(2v_(0))sin alpha=((2v_(0))sqrt(4w^(2)-{w+(sqrt3-1)y}^(2)))/(2w)`
`=v_(0)/wsqrt(4w^(2)-{w+(sqrt3-1)y}^(2))`
or `int_(0)^(w)=(dy)/sqrt(4w^(2)-{w+(sqrt3-1)y}^(2))=v_(0)/w int_(0)^(t)dt`
Solving this, we get `t=(wpi)/(6(sqrt3-1)v_(0))`