Question

Two cars $A$ and $B$ are $5\text{km}$ apart on a line joining South to North. Car $A$ further North is streaming west at $10\text{km/hr}$ and Car $B$ is streaming North at $10\text{km/hr}$. What is their closest distance of approach?

• A. $5\text{km}$
• B. $\frac{5}{\sqrt{2}}\text{km}$
• C. $2\text{km}$
• D. $\sqrt{3}\text{km}$

Answer 1

The correct option is B $\frac{5}{\sqrt{2}}\text{km}$

Velocity of car A, $\overrightarrow{V_A}=-10\hat{j}\text{km/h}$
Velocity of car B, $\overrightarrow{V_B}=10\hat{i}\text{km/h}$
Velocity of car B w.r.t A, $\overrightarrow{V_{BA}}=\overrightarrow{V_B}-\overrightarrow{V_A}=10\hat{i}+10\hat{j}$
Magnitude, $\overrightarrow{V_{BA}}=\sqrt{{10}^2+{10}^2}=10\sqrt{2}\text{km/h}$
$\tan \theta =\frac{10}{10}=1\Rightarrow \theta ={45}^{\circ }$
Talking w.r.t ${}^{\prime }{A}^{\prime }$

For minimum distance,
$S_{min}=AC=AB\sin 45°$
$=5\left(\frac{1}{\sqrt{2}}\right)$
$=\frac{5}{\sqrt{2}}\text{km}$