Question

Boat A is sailing towards north-east with velocity $\overrightarrow{v}=5\hat{i}+5\hat{j}km/hr$, where $\hat{i}$ points east and $\hat{j}$, north. Boat B is at a distance of $50km$ east and $100km$ north of boat A and is sailing towards west at $10km/hr$. A will be at minimum distance from B in:

• A. $2hrs$
• B. $3hrs$
• C. $4hrs$
• D. $5hrs$

Answer 1

The correct option is D $5hrs$


If we take the position of boat 'A' as origin then the positions and velocities of both ships can be given as:
$\overrightarrow{v_A}=5\hat{i}+5\hat{j}km/hr$
$\overrightarrow{v_B}=-10\hat{i}km/hr$
$\overrightarrow{r_A}=(0\hat{i}+0\hat{j})km$, $\overrightarrow{r_B}=(50\hat{i}+100\hat{j})km$

Time after which distance between them will be minimum
$t=-\frac{\overrightarrow{r_{BA}}.\overrightarrow{v_{BA}}}{|\overrightarrow{v_{BA}}{|}^2}$;
Where $\overrightarrow{r_{BA}}=50\hat{i}+100\hat{j}km$
$\overrightarrow{v_{BA}}=-10\hat{i}-(5\hat{i}+5\hat{j})=(-15\hat{i}-5\hat{j})km/hr$
$\therefore t=-\frac{|(50\hat{i}+100\hat{j}).(-15\hat{i}-5\hat{j})|}{|-15\hat{i}-5\hat{j}{|}^2}$
$=\frac{750+500}{250}hrs=\frac{1250}{250}hrs=5hrs$