Question

The speed of a boat in still water is $\sqrt{5}\text{m/s}$ . While crossing a river, the average speed of the boat is $3\text{m/s}$. If boat crosses the river in the shortest possible time, what is the speed of flow of water?

• A. $2\text{m/s}$
• B. $4\text{m/s}$
• C. $6\text{m/s}$
• D. $8\text{m/s}$

Answer 1

The correct option is A $2\text{m/s}$

To cross the river by the shortest time, one has to row the boat perpendicular to the river.
Let $\overrightarrow{V_{br}}$ be the velocity of boat with respect to river. $\overrightarrow{V_b}$ and $\overrightarrow{V_r}$ are the velocities of boat and river w.r.t ground respectively.

According to question,
$|\overrightarrow{V_{br}}|=\sqrt{5}\text{m/s}$ and $|\overrightarrow{V_b}|=3\text{m/s}$
We know that
$\overrightarrow{V_{br}}=\overrightarrow{V_b}-\overrightarrow{V_r}$
$\overrightarrow{V_b}=\overrightarrow{V_{br}}+\overrightarrow{V_r}$
$|\overrightarrow{V_b}|=\sqrt{V_{br}^2+V_r^2}$
$\Rightarrow V_r^2+5=9$
$\Rightarrow V_r=2\text{m/s}$

Alternative:
Use following equation
$t=\frac{AB(=d)}{V_{br}}=\frac{BC}{V_r}=\frac{AC}{V_b(=\sqrt{V_{br}^2+V_r^2})}$
$|\overrightarrow{V_b}|=\sqrt{V_{br}^2+V_r^2}$
$\Rightarrow V_r^2+5=9$
$\Rightarrow V_r=2\text{m/s}$