A particle, $$\(P\)$$, of mass $$\(0.4 \mathrm{~kg}\)$$ is attached to the midpoint of a light elastic spring of natural length $$\(0.8 \mathrm{~m}\)$$ and modulus of elasticity $$\(20 \mathrm{~N}\)$$. The ends of the spring are attached to the fixed points $$\(A\)$$ and $$\(B\)$$ on a smooth horizontal table, where $$\(A B=1.2 \mathrm{~m}\)$$. Initially $$\(P\)$$ is at rest at the midpoint $$\(O\)$$ of $$\(A B\)$$ where $$\(A O B\)$$ is a straight line. The particle $$\(P\)$$ now receives an impulse of magnitude $$\(2 \mathrm{Ns}\)$$ so that $$\(P\)$$ starts to move directly towards $$\(B\)$$. (a) Prove that $$\(P\)$$ moves with simple harmonic motion. (4) (b) Write down, in terms of $$\(\pi\)$$, the period of the motion. (1) (c) Find the amplitude of the motion. (3) (d) Find the length of time in each complete oscillation for which $$\(A P\)$$ is greater than $$\(0.5 \mathrm{~m}\)$$. (5)

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