River flows with a velocity u along its path. Let the boat have a uniform velocity v with respect to still water.
Boat wants to cross from point A on one side of the river to any point B on the other side of the river. We want the boat to take minimum time duration.
Let the boat travel at an angle of Ф wrt to u ie. flow of river. At any time the velocity of the boat has two components:
along the river flow = u + v cos Ф
perpendicular to river flow = v sin Ф
The velocities are uniform and depend only on the angle Ф.
If the width of the river (assume to be uniform) = d, then, the time duration to travel to the other side = t = d / v sinФ
v sin Ф is the component of velocity along the path d.
Hence, the time duration is minimum if the angle Ф = 90°.