Show that for an isolated system the centre of mass moves with a uniform velocity along a straight line path.
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Let M be the total mass of a system supposed to be concentrated at the centre of mass whose position vector is $\vec{\text{r}}$ then in the presence of an external force $\vec{\text{F}},$we have $\vec{\text{F}}=\text{M}\frac{\text{d}^2\vec{\text{r}}}{\text{dt}^2}=\text{M}\frac{\text{d}}{\text{dt}}\Big(\frac{\overrightarrow{\text{dr}}}{\text{dt}}\Big)$ $=\text{M}\frac{\text{d}}{\text{dt}}(\vec{\text{v}}_{\text{cm}})$ However for an isolated system, force $\vec{\text{F}}=0$ and hence, we have $\text{M}\frac{\text{d}}{\text{dt}}(\vec{\text{v}}_{\text{cm}})=0$ $\text{or }\frac{\text{d}}{\text{dt}}(\vec{\text{v}}_{\text{cm}})=0$ $\text{or }\vec{\text{v}_{\text{cm}}}=\text{a}\text{ constant}$ It means that for an isolated system the centre of mass moves with a uniform velocity along a straight line path.
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