A boat moves in still water with a velocity which is `k` times less than the river flow velocity. Find the angle to the stream direction at which the boat should be rowed to minimize drifting.

Answer

`cos^(-1)(1//k)`

Explanation

Let the flow velocity of river be `u` and the velocity of boat in still water be `v`. Thus,
`" "v = (u) /(K)`

Also, let the boat move at an angle `theta` with the direction of stream.
Now the velocity of boat in the river is the vector resultant of the velocity of boat and flow velocity of river, which can be writtenas
`" "vecv_(B) = (u-vcosalpha) hati+ (usin alpha) hatj`
`" "= (u+ vcos theta) hati+ (u sin theta) hatj`
Hence, the time taken to cross the river `= (d)/(usin theta)`
(d= width of the river)
Thus, the drift `s= (u + vcostheta)*d`
or `" "s= d(cosectheta + (v)/(u)cottheta)`
or `" "(ds)/(dtheta)= d(cosectheta cottheta- (v)/(mu)cosec^(2)theta)=0`
or `" "(v)/(u) cosec^(2) theta = cosectheta cottheta`
or `" "costheta = (1)/(k) or theta = cos^(-1)((1)/(k))`