Question

A boat can go across a lake and return in time $T_0$ at a speed $v$. On a rough day there is a uniform current at speed $v_1$ to help the onward journey and impede the return journey. If the time taken to go across and return on the same day be $T$, then $T/T_0$ will be

• A. $\frac{1}{(1-v_1^2/v^2)}$
• B. $\frac{1}{(1+v_1^2/v^2)}$
• C. $(1-v_1^2/v^2)$
• D. $(1+\frac{v_1^2}{v_2})$

Answer 1

The correct option is A $\frac{1}{(1-v_1^2/v^2)}$

Let $d$ be the width of lake to be crossed, then
$T_0=\frac{2d}{v}$
and $T=\frac{d}{(v+v_1)}-\frac{d}{(v-v_1)}=\frac{2dv}{v^2-v_1^2}$
$=\frac{2dv}{v^2(1-v_1^2/v^2)}=\frac{2d}{v(1-v_1^2/v^2)}$
So, $\frac{T}{T_0}=\frac{1}{(1-v_1^2/v^2)}$.