Question

Ship $A$ is sailing towards north-east with velocity) $\overset{ \rightarrow }{v}=30\hat{i}+50\hat{j}kmh^{- 1} \, $ where $\hat{i}$ points east and $\hat{j},$ north. The ship $B$ is at a distance of $80 \, km$ east and $150 \, km$ north of Ship $A$ and is sailing towards the west at $10kmh^{- 1}.$ $A$ will be at the minimum distance from $B$ in:

• A. $4.2 \, h$
• B. $ \, 3.2 \, h$
• C. $ \, 2.6 \, h$
• D. $2.2 \, h$

Answer 1

Answer: (C)

$ \vec{v}_{r e l} =\vec{v}_1-v_2=40 i+50 j $
$ \vec{r}_{r e l} =\vec{r}_1-\vec{r}_2=-80 i-150 j $
$ t_{\min } =\frac{\vec{v}_{r e l} \cdot \vec{r}_{r e l}}{\vec{v}_{r e l}^2}=\frac{-3200-7500}{{\sqrt{40^2+50^2}}^2} $
$ t_{\min } =\frac{10700}{4100}=\frac{107}{41}=2.6\, h $