Question

Ship $A$ is sailing towards north $-$ east with velocity $\overrightarrow{v}=(30\hat{i}+50\hat{j})km/hr$ where $\hat{i}$ points east and $\hat{j}$ north. Ship $B$ is at a distance of $80km$ east and $150km$ north of Ship $A$ and is sailing towards west at $10km/hr$. $A$ will be at minimum distance from $B$ in

• A. $3.2hr$
• B. $2.2hr$
• C. $4.2hr$
• D. $2.6hr$

Answer 1

The correct option is D $2.6hr$

1. Draw rough figure for the given situtation.

If we take the position of ship ${}^{\prime }{A}^{\prime }$ as origin, then positions and velocities of both ships can be drawn as

2. Find position and velocity of $B$ with respect to $A$

Position of ship $A,{\overrightarrow{r}}_A=0\hat{i}+0\hat{j}$

And position of ship $B$,

${\overrightarrow{r}}_B=(80\hat{i}+150\hat{j})km$

position of ship $B$ with respect to $A$

${\overrightarrow{r}}_{BA}={\overrightarrow{r}}_B-{\overrightarrow{r}}_A$

${\overrightarrow{r}}_{BA}=(80\hat{i}+150\hat{j})km$

Velocity of ship $A$,

${\overrightarrow{v}}_A=(30\hat{i}+50\hat{j})km/hr$

And velocity of ship $B,{\overrightarrow{v}}_B=-10\hat{i}km/hr)$

Velocity of ship $B$ with respect to $A$

${\overrightarrow{v}}_{BA}={\overrightarrow{v}}_B-{\overrightarrow{v}}_A$

${\overrightarrow{v}}_{BA}=-10\hat{i}-(30\hat{i}+50\hat{j})$

${\overrightarrow{v}}_{BA}=(-40\hat{i}-50\hat{j})km/hr$

3. Find the time after which distance between ships will be minimum.

Time after which diatance between them wil be minimum

$t=\frac{{\overrightarrow{r}}_{BA}.{\overrightarrow{v}}_{BA}}{{\left|{\overrightarrow{v}}_{BA}\right|}^2}$

$t=\frac{(80\hat{i}+150\hat{j}).(-40\hat{i}-50\hat{j})}{{|-40\hat{i}-50\hat{j}|}^2}$

$t=\frac{3200+7500}{4100}hr$

$t=2.6hr$

Final answer: (d)