Simple harmonic motion from a sine displacement, the velocity–displacement relation, and the density of the displacement

A particle oscillates along a horizontal straight line between two end points, and the fixed point \(O\) lies midway between them. At time \(t\) seconds, the displacement of the particle from \(O\) is \(x\) metres, where \(x=4\sin(2t+\frac\pi3)\) and \(0\leq t\leq\pi\).

1. Determine
   1. the amplitude of the motion;[1]
   2. the displacement of the particle when \(t=0\);[2]
   3. the time at which the particle reaches \(O\) for the first time.[2]
2. Consider, more generally, a particle whose acceleration \(a\) and displacement \(s\) are related by \(a=-n^2s\), where \(n>0\).
   At time \(t\), its displacement is \(s=A\sin(nt+b)\), where \(t\geq0\), \(A>0\), \(n>0\) and \(-\pi\leq b\leq\pi\).
   Differentiate \(s\) to obtain \(\frac{\mathrm{d}s}{\mathrm{d}t}\) and \(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\), and hence verify that \(s=A\sin(nt+b)\) satisfies \(a=-n^2s\).[2]
3. 1. Using the chain rule, show that \(a=v\frac{\mathrm{d}v}{\mathrm{d}s}\), where \(v\) denotes the velocity.[1]
   2. Solve the differential equation \(v\frac{\mathrm{d}v}{\mathrm{d}s}=-n^2s\) to show that \(v^2=n^2(A^2-s^2)\).[5]
   3. Hence, or otherwise, find the greatest speed of the particle.[2]
4. Let the continuous random variable \(S\) be the displacement \(s\) of the particle from \(O\) at time \(t\). The probability density function of \(S\) is
   \[f(s)=\begin{cases}\frac1{\pi\sqrt{A^2-s^2}},&-A<s<A,\\0,&\text{otherwise}.\end{cases}\]
   Show that \(P(-\frac A2\leq S\leq\frac A2)=\frac13\).[4]
5. For \(-A<s<A\), \(f(s)\) may be written as \(\frac{k}{|v(s)|}\), where \(k\) is a positive constant and \(v(s)\) is the velocity of the particle when its displacement from \(O\) is \(s\).
   Find \(k\).[3]
6. 1. Determine \(E(S)\), giving a reason.[2]
   2. Interpret this value in terms of the motion of the particle.[1]