4 A particle P is performing simple harmonic motion in a vertical line. At time \(t \mathrm {~s}\), its displacement \(x \mathrm {~m}\) above a fixed point O is given by $$x = A \sin \omega t + B \cos \omega t$$ where \(A , B\) and \(\omega\) are constants.

1. Show that the acceleration of P , in \(\mathrm { ms } ^ { - 2 }\), is \(- \omega ^ { 2 } x\). When \(t = 0 , \mathrm { P }\) is 16 m below O , moving with velocity \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards, and has acceleration \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards.
2. Find the values of \(A , B\) and \(\omega\).
3. Find the maximum displacement, the maximum speed, and the maximum acceleration of P .
4. Find the speed and the direction of motion of P when \(t = 15\).
5. Find the distance travelled by P between \(t = 0\) and \(t = 15\).