# Fatal attraction

**Electricity and Magnetism**Level 5

Consider a universe where gravity is proportional to the distance between two masses. That is to say, any mass \(m_{i}\) attracts any other mass \(m_{j}\) with a force given by

\[\vec{ F}_{ij}=G m_{i}m_{j} (\vec{r}_{i}-\vec{r}_{j}) \]
with \(G=1\times 10^{-34}~\mbox{N}/(\mbox{m}\cdot \mbox{kg}^{2})\).
Here \(\vec{r}_{i}\) and \(\vec{r}_{j}\) are the positions of the masses \(m_{i}\) and \(m_{j}\). In this odd universe, an isolated star with mass \(M= 1\times 10^{30 }~\textrm{kg} \) explodes into many fragments. It turns out that after some time \(\tau \) all the fragments coalesce, that is, they meet at a single point even if the explosion is anisotropic. Find the time \(\tau\) **in seconds**. Assume that Newton's law holds true in this universe and that the star was initially at rest.

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