A steamer plies between two stations `A and B` on opposite banks of a river, always following the path `AB`. The distance between stations `A and B` is `1200 m`. The velocity of the current is `sqrt(17) m s^-1` and is constant over the entire width of the river. The line `AB` makes an angle `60^@` with the direction of the flow. The steamer takes `5 min` to cover the distance `AB` and back.
(a) Find the velocity of steamer with respect to water.
(b) In what direction should the steamer move with respect to line `AB` ?

Answer

`tan^-1 sqrt((3)/(17)) ; sqrt(85) ms^-1`

Explanation

From A to B : `AB = (v cos beta + u cos prop) t_1`
From B to A : `AB = (u cos beta - u cos prop)t_2`
`prop = 60^@, v sin beta = u sin prop`...(i)
`rArr 5 xx 60 = (1200)/(v cos beta + u cos prop) + (1200)/((v cos beta - v cos prop))`
Putting `u = sqrt(17), prop = 60^@`, we get
`v^2 cos^2 beta - 8v cos beta -(17)/(4) = 0 rArr v cos beta = (17)/(2)` ...(ii)
From Eqs. (i) and (ii), `tan beta = sqrt((3)/(17))`
`rArr beta = tan^-1 sqrt((3)/(17)) and v = (sqrt(85)) ms^-1`.