Question

A man wants to cross a river (from point A to B) 500 m wide. Rowing speed of the man relative to water is 3 km/h and river flows at the speed of 2 km/hr. If man’s walking speed on the shore is 5 km/hr, then at what angle with line AB he should start rowing in order to reach the directly opposite point on the other bank in shortest time.

• A. $si{n}^{-1}\left(\frac{4}{7}\right)$
• B. $co{s}^{-1}\left(\frac{3}{7}\right)$
• C. $si{n}^{-1}\left(\frac{3}{7}\right)$
• D. $co{s}^{-1}\left(\frac{4}{7}\right)$

Answer 1

The correct option is C $si{n}^{-1}\left(\frac{3}{7}\right)$


${\overrightarrow{V}}_{MG}=(u-vsin\theta )\hat{i}+vcos\theta \hat{j}$
time taken to cross the river, $t_1=\frac{0.5}{vcos\theta }$
drift (d) = $(u-vsin\theta )\frac{0.5}{vcos\theta }$
time to cover this distance, $t_2=\frac{d}{5}$
total time T = $t_1+t_2$
for T to be minimum, $\frac{dT}{d\theta }=0$
solving $\theta =si{n}^{-1}\left(\frac{3}{7}\right)$