2. A particle \(P\) moves along the \(x\)-axis such that its position \(x\) metres, after \(t\) seconds, is given by
$$x = \sin ( \pi t ) + \sqrt { 3 } \cos ( \pi t )$$
- Show that the motion of the particle \(P\) is Simple Harmonic. State the value of \(x\) at the centre of motion.
- Show that the period of the motion of \(P\) is 2 s and determine the amplitude.
Suppose that another particle \(Q\) is introduced so that it also moves along the \(x\)-axis with Simple Harmonic Motion with centre of motion, \(O\), and period equal to that of particle \(P\). When \(t = 0\), the particle \(Q\) is at \(O\) and when it is \(2 \sqrt { 3 } \mathrm {~m}\) from \(O\) its speed is \(2 \pi \mathrm {~ms} ^ { - 1 }\).
- Find the amplitude of particle \(Q\).
- Determine the time when particles \(P\) and \(Q\) first meet.