Show that a boat must move at an angle with respect to river water in order to cross the river in minimum time .

Explanation

Motion of a boat in a river : Let a boat can travel with a speed of `V_(bE)` in still water w.r. to earth . It is used to cross a river which flows with a speed of `V_(WE)` with respect to earth . Let width of river is W .
We can cross the river in two different ways . 1) in shortest path 2) in shortest time .
To cross the river in shortest time :
Shortest time ,t `= (" width of river")/("velocity of boat") = (W)/(V_(bE))`
To cross the river in shortest time boat must be rowed along the width of river i.e., boat must be rowed perpendiculat to the bank of `90^(@)` with the flow of water . because width of river is the shortest distance . so velocity must be taken in that direction to obtain shortest time . In this case `V_(bE)` and `V_(WE)` are perpendicular and boat will travel along AC . The distance BC is called drift . So to cross the river in shortest time angle with flow of water ` = 90^(@)` .