# Ship A is sailing towards north-east with velocity $\vec{v}= 30 \widehat{i}+50\widehat{j}$ km/hr where $\widehat{i}$ points east and $\widehat{j}$, north. Ship B is at a distance of 80 km east and 150 km north of Ship A and is sailing towards west at 10 km/hr. A will be at minimum distance from B in :Option 1)2.6 hrs.Option 2)2.2 hrs.Option 3)4.2 hrs.Option 4)3.2 hrs.

$v_{A}=30\widehat{i}+50\widehat{j}$

$r_{A}=0\widehat{i}+0\widehat{j}$

$r_{B}=80\widehat{i}+150\widehat{j}$

$v_{B}=-10\widehat{i}$

$\vec{v_{r}}=40\widehat{i}+50\widehat{j}$

$\vec{r_{r}}=-80\widehat{i} - 150\widehat{j}$

$t_{min}=\frac{\left | \overrightarrow{v_{r}}.\overrightarrow{r_{r}} \right |}{\left | \overrightarrow{v_{r}} \right |}=\frac{(40\times 80+150 \times50)}{4100}$

$t_{min}= \frac{10700}{4100}=2.6\; hrs.$

Option 1)

2.6 hrs.

Option 2)

2.2 hrs.

Option 3)

4.2 hrs.

Option 4)

3.2 hrs.

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