Problem

A particle $P$ is moving in a straight horizontal line. At time $t\,\mathrm{s}$, the displacement of $P$ from a fixed point $O$ on the line is $x\,\mathrm{m}$ and the velocity of $P$ is $v\,\mathrm{m\,s^{-1}}$. The acceleration of $P$ is $\frac12(v^2+4)\,\mathrm{m\,s^{-2}}$ in the direction $PO$. Initially $P$ is at $O$ and is moving with velocity $2\,\mathrm{m\,s^{-1}}$.

(a) Find an expression for $x$ in terms of $t$.

(b) Find the time when $P$ next goes through $O$.