Answer to Question #110611 in Calculus for Chinmoy Kumar Bera

Question

two boat each move a velocity v relative to the water and both cross a river of breadth a running with uniform velocity V.they start together one boat crossing by shortest path and the otherin shortest time.find the difference between the time of arrival.

Answer 1

Shortest path

Relative Velocity of boat is in a direction at an angle $\theta$ with the vertical

$vsin\theta=V$

$cos\theta=\sqrt{\frac{v^2-V^2}{v^2}}$

$Time=\frac{Distance}{speed}$

$t_1=d/(vcos\theta)$

$t_1=\frac{d}{\sqrt{v^2-V^2}}$

Shortest time

Relative velocity of boat is perpendicular to the velocity of river.

$t_2=d/v$

$t_2-t_1=d[\frac1 v-\frac{1}{\sqrt{v^2-V^2}}]$