Question

A boat capable of running with velocity $v$ relative to water, in a flowing river having velocity as $u$. The boat travels a distance $d$ downstream and returns back to the original position. Find out the time taken in a complete trip.

• A. $\frac{2dv}{\sqrt{v^2-u^2}}$
• B. $\frac{2dv}{v^2-u^2}$
• C. $\frac{2dv}{v^2+u^2}$
• D. $\frac{\sqrt{2}dv}{v^2-u^2}$

Answer 1

The correct option is B $\frac{2dv}{v^2-u^2}$

Velocity of river flow $v_r=u$
velocity of boat w.r.t river $v_{b,r}=v$
$v_{b,r}=v_b-v_r...\left(i\right)$
$v_b$ is the velocity of boat w.r.t ground.
during downstream journey,
$v_b=v_{b,r}+v_r$
$\Rightarrow v_b=v+u$
hence time taken to complete the distance $d$ is,
$t_1=\frac{d}{v_b}=\frac{d}{v+u}...\left(ii\right)$
during Upstream journey, (taking flow of river as $-ve$ direction).
$v_b=v_{b,r}+v_r$
$\Rightarrow v_b=v+(-u)=v-u$
hence time taken to complete the distance $d$ is,
$t_2=\frac{d}{v_b}=\frac{d}{v-u}...\left(iii\right)$


Total time taken in a round trip is,
$t=t_1+t_2$
$t=\frac{d}{v+u}+\frac{d}{v-u}$
$t=\frac{d(v-u)+d(v+u)}{(v+u)(v-u)}$
$\therefore t=\frac{2dv}{v^2-u^2}$