Question

Sol. [Key Concept: $v_{cm}$ and $a_{cm}$ come out parallel ($v_{cm} = 2i + 2j$, $a_{cm} = 3i + 3j$), so the centre of mass moves in a straight line.]

$${{\overrightarrow{v}}_{cm} = \frac{m_{1}{\overrightarrow{v}}_{1} + m_{2}{\overrightarrow{v}}_{2}}{m_{1} + m_{2}} = \frac{m(4\widehat{i}) + m(4\widehat{j})}{2m} = 2\widehat{i} + 2\widehat{j} }{{\overrightarrow{a}}_{cm} = \frac{m_{1}{\overrightarrow{a}}_{1} + m_{2}{\overrightarrow{a}}_{2}}{m_{1} + m_{2}} = \frac{m(6\widehat{i} + 6\widehat{j}) + m(0)}{2m} = 3\widehat{i} + 3\widehat{j} - }{{\overrightarrow{v}}_{cm}\|{\overrightarrow{a}}_{cm} }$$so, path is straight line