A man can row a boat with 4 km/h in still water. If he is crossing a river where the current is 2 km/h.
(a) In what direction will his boat be headed if he wants to reach a point on the other bank, directly opposite to starting point ?
(b) If width of the river isa 4 km, how long will the man take to cross the river, with the condition in part (a) ?
(c ) In what direction should he head the boat if he wants to cross the river in shortest time and what is this minimum time ?
(d) How long will it take him to row 2 km up the stream and then back to his starting point ?

Explanation

(a) Velocity of a boat on still water is its capacity to move on water surface and aquals to its velocity relative to water.
`vec(v)_(b//w)=` Velocity of boat relative to water = Velocity of boat on still water

On flowing water, the water carries the boat along with it. Thus velocity `vec(v)_(b)` of the boat relative to the ground equals to vector sum of `vec(v)_(b//w)` and `vec(v)_(w)`. The boat crosses the river with the velocity `vec(v)_(b)`.
`vec(v)_(b)=vec(v)_(b//w)+vec(v)_(w)`
(b) To cross the river perpendicular to current the boat must be steered in a direction so that one of the components of its velocity `(vec(v)_(b//w))` relative to water becomes equal and opposite to water flow velocity `vec(v)_(w)` to neutralize its effect. It is possible only when velocity of boat relative to water is greater than water flow velocity. In the adjoining figure it is shown that the boat starts from the point O and moves along the line OP (y-axis) due north relative to ground with velocity `vec(v)_(b)`. To achieve this it is steered at an anglr `theta` with the y-axis.
`v_(b//w) sin theta=v_(w) rarr 5 sin theta=3 rArr theta=37^(@)`

(c) The boat will cover river width b with velocity
`v_(b)=v_(b//wy)=v_(b//w) sin 37^(@)=4 m//s` in time t, which is given by
`t=b//v_(b) rarr t=50 s`
(d) To cross the river in minimum time, the component perpendicular to current of its velocity relative to ground must be kept to maximum value. It is achieved by steering the boat always perpendicular to current as shown in the adjoining figure. The boat starts from O at the south bank and reaches point P on the north bank. Time t taken by the boat is given by
`t=b//v_(b//w) rarr t=40 s`
Drift is the displacement along the river current measured from the starting point. Thus, it is given by the following equation. We denote it by `x_(d)`.
`x_(d)=v_(bx)t`
Substituting `v_(bx)=v_(w)=3 m//s`, from the figure, we have `x_(d)=120 m`