Question

To cross the river in shortest distance, a swimmer should swim making an angle θθ with the upstream. What is the ratio of the time taken to swim across in the shortest time to that in swimming across over shortest distance. [Assume that the speed of swimmer in still water is greater than the speed of river flow]

Answer 1

$t_1=\frac{d}{v_{s}r}$

$V_{s}rsin\theta =V_R$

$\frac{t_1}{t_2}=\frac{\sqrt{V_s{R}^2-{VR}^2}}{V_s{R}^2}$

$=\frac{\sqrt{1-{VR}^2}}{V_s{R}^2}$

$=\frac{d}{V_{s}Rcos\theta }$

$t_2=\frac{d}{\sqrt{V_s{R}^2-{VR}^2}}$