A particle \(P\) moves in a straight line so that, after \(t\) seconds, its distance, \(s\) metres, from a fixed point \(O\) is given by the equation. \[s=12\sin\left(\frac{3}{2}t\right)\] Determine

  1. expressions for the velocity and acceleration of \(P\) in terms of \(t\),

  2. the time at which \(P\) first returns to \(O\),

  3. the maximum distance of \(P\) from \(O\) during the motion,

  4. the velocity of \(P\) when \(s=6\).