A man crosses a river in a boat. If he cross the river in minimum time he takes `10 min` with a drift `120 m.` If he crosses the river taking shortest path, he takes `12.5 min,` find
(a) width of the river
(b) velocity of the boat with respect to water
(c) speed of the current

Explanation

Case I : If the man crosses the river in minimum time he should move perpendicular to bank or normal to the direction of water flow.
Let `|vec v_(m,w)| = v and | vec v_w| = u`
Time to cross river, `t_1 = 10 = (d)/(v) rArr d = 10 v`,
`x = ut_1 rArr 120 u = u xx 10 rArr u = 12 m min^-1`
.
Case 2 : If the man crosses river taking the shortest route, the drift should be zero. Time to cross river.
`t_2 = 12.5 = (d)/(sqrt(v^2 - u^2))`
From case 1, `d = 10 v and v = 12 m s^-1`
`12.5 = (10 v)/(sqrt(v^2 - 12^2)) rArr v = 20 ms^-1`
.