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

A boat moves relative to water with a velocity which is n times less t

1.	A boat moves relative to water with a velocity which is n times less than the river flow velocity . At what angle to the stream direction must the boat move to minimize drifting?
Answer» Solution :In this problem, one thing should be carefully noted that the velocity 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 starts at an angle `theta` from the normal directionup stream as shown. now, again if we find the components of velocity of boat along and perpendicular to the flow , these are, velocity along the river, `V_(x)=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/(v cos theta)` In this time, drift `x=(v_(x))t=(u-vsintheta)d/(vcostheta)"(or)" x=(ud)/v SEC theta - d tan theta`. The drift x is minimum , when `(dx)/(dtheta)=0"(or)" ((ud)/v) (sec theta. tan theta) - d sec^(2) theta =0` (or)`u/vsin 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 DIRECTION or an angle `pi/2+sin^(-1)(v/u)` from stream direction.