Question

A boat can go across a lake and return in time $t_0$ at a speed $u$. On a rough day, there is uniform current at speed $v$ to help the onward journey and impede the return journey. If the time taken to go across and return on the rough day be $t$, then $\frac{t}{t_0}$ is equal to

• A. $1-\frac{v^2}{u^2}$
• B. $\frac{1}{1-\left(\frac{v^2}{u^2}\right)}$
• C. $1+\frac{v^2}{u^2}$
• D. $\frac{1}{1+\left(\frac{v^2}{u^2}\right)}$

Answer 1

Answer: (B)

$\frac{1}{1-\left(\frac{v^2}{u^2}\right)}$

Let $L$ be the length of lake so, for normal day time $t_o=\frac{2L}{u}$

and for rough day time $t=\frac{L}{u+v}+\frac{L}{u-v}=\frac{2Lu}{u^2-v^2}$

therefore $\frac{t}{t_o}=\frac{u^2}{u^2-v^2}=\frac{1}{1-\left(\frac{v^2}{u^2}\right)}$