A man in a row boat must get from point `A` to point `B` on the opposite bank of the river (see figure).The distance `BC=a`.The width of the river `AC=b`.At what minimum speed `u` relative to the still water should the boat travel to reach the point `B`?The velocity of flow of the river is `v_(0)`

Options

A. `(v_(0)b)/sqrt(a^(2)+b^(2))`
B. `(v_(0)a)/sqrt(a^(2)+b^(2))`
C. `(v_(0)b)/(sqrt2a)`
D. `(v_(0)a)/(sqrt2b)`

Answer

A

Explanation

Suppose `u` is the speed of the boat relative to water, then velocity of the flow (w.r.t bank) `V_(0)`
`v_(x)=(u cos theta+v_(0))`, and perpendicular to flow will be `v_(y)=u sin theta`. Time to cross the river,`t=b/(u sin theta)`.In the time the distance travelled by the boat in the direction of flow
`a=v_(x)t=(ucostheta+v_(0))b/(u sin theta)`
or `ausin theta=ub cos theta+v_(0)b`
`:.u=(v_(0)b)/((a sin theta-b cos theta))`..(i) u to be minimum duld
`theta=0 or d/(d theta)[(v_(0)b)/(a sin theta-b cos theta)]=0`
or ` tan theta=-a/b :. cos theta=b/sqrt(a^(2)+b^(2))`
On substituting these values in equation (i), we get
`u_(min)=(V_(0)b)/sqrt(a^(2)+b^(2))`.