A particle \(P\) moves on a straight line in simple harmonic motion. The centre of the motion is \(O\). The points \(A\) and \(B\) are on the line, on opposite sides of \(O\), with \(O A=1.6 \mathrm{~m}\) and \(O B=1.2 \mathrm{~m}\). The ratio of the speed of \(P\) at \(A\) to its speed at \(B\) is \(3: 4\).
(i) Find the amplitude of the motion.

The maximum speed of \(P\) during its motion is \(\frac{1}{3} \pi \mathrm{~m} \mathrm{~s}^{-1}\).
(ii) Find the period of the motion.

(iii) Find the time taken for \(P\) to travel directly from \(A\) to \(B\).