Question

Ship $A$ is located $4\text{km}$ north and $3\text{km}$
east of ship $B$. Ship $A$ has a velocity of $20{\text{km h}}^{-1}$ towards the south and ship $B$ is moving at $40{\text{km h}}^{-1}$ in a direction ${37}^{\circ }$ north of east. $X$ and $Y$ axes are along east and north directions, respectively

• A. Velocity of $A$ relative to $B$ is $-32\hat{i}-44\hat{j}$
• B. Position of $A$ relative to $B$ as a function of time is given by
  ${\overrightarrow{r}}_{AB}=(3-32t)\hat{i}+(4-44t)\hat{j}$
• C. Velocity of $A$ relative to $B$ is $32\hat{i}-44\hat{j}$
• D. Position of $A$ relative to $B$ as a function of time is given by $(32t\hat{i}-44t\hat{j})$

Answer 1

Answer: (A), (B)

Position of $A$ relative to $B$ as a function of time is given by

${\overrightarrow{r}}_{AB}=(3-32t)\hat{i}+(4-44t)\hat{j}$


${\overrightarrow{v}}_A=-20\hat{j}{\text{kmh}}^{-1}$
${\overrightarrow{v}}_B=\left[\right(40\cos {37}^{\circ })\hat{i}+(40\sin {37}^{\circ }\left)\hat{j}\right]{\text{kmh}}^{-1}$
$=(32\hat{i}+24\hat{j}){\text{kmh}}^{-1}$
${\overrightarrow{v}}_{AB}={\overrightarrow{v}}_A-{\overrightarrow{v}}_B$
$=(-20\hat{j})-(32\hat{i}+24\hat{j})$
$=-32\hat{i}-44\hat{j}$
Option (a) is correct
Option (c) is incorrect
At any time
${\overrightarrow{S}}_A=3\hat{i}+4\hat{j}+(-20\hat{j})t$
${\overrightarrow{S}}_B=(32\hat{i}+24\hat{j})t$

Position of $A$ relative to $B$:
${\overrightarrow{S}}_{AB}={\overrightarrow{S}}_A-{\overrightarrow{S}}_B$
$=(3\hat{i}+4\hat{j}-20t\hat{j})-(32t\hat{i}+24t\hat{j})$
$=(3-32t)\hat{i}+(4-44t)\hat{j}$
Option (b) is correct.
Option (d) is incorrect.