A boat moves relative to water with a velocity with a velocity v is n times less than the river flow u. At what angle to the stream direction must the boat move to minimize drifting ?

Explanation

in this problem . One thing should be carefully noted of boat is less than the river flow velocity . In such a case , boat cannot reach the point directly opposite to its starting point i.e., drift can never be zero , thus to minimize the drift boat start at an angle ` theta ` from normal direction up sterm as shown

Now again if we find the components of velocity of boat and perpendicular to the flow these are velocity along the river ` V_t = u - V sin theta`
and velocity perpendicular to the river `,V_y = v cos theta `
time taken to cross the river is ` t= ( d )/( v_y) = (d )/( cos theta)`
In this time drift ` x=(v_A) t= ( u -v sin theta) (d)/( v_y) =(d)/( v cos theta)`
in this time drift `x =( v_x) t= ( u - V sin theta ) (d)/( v cos theta) (or) x= (ud )/( v ) sec theta -d tan theta `
the drift x is minimum when `( dx )/(d theta ) =0 (or) ( (ud)/(v)) ( sec theta , tan theta) - d sec^2 theta =0`
` (or) (a)/(v ) sin theta =1 (or) sin theta = ( v )/( u ) = (1 )/(n) ( as v=(u )/(n))`
so for minimum drift the boat must move at an angle ` theta = sin ^(-1) ((v)/( u ))` from normal direaction or an angle `(pi)/(2) + sin ^(-1) ((v)/(u ))` from steam direction.