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 \(v \mathrm{~ms}^{-1}\). The acceleration of \(P\) is \(\frac{1}{2}\left(v^{2}+4\right) \mathrm{ms}^{-2}\) in the direction \(P O\). Initially \(P\) is at \(O\) and is moving with velocity \(2 \mathrm{~ms}^{-1}\). (a) Find an expression for \(x\) in terms of \(t\). (b) Find the time when \(P\) next goes through \(O\).