IB AA HL May 2026 P3 TZC Q2 (H2 adaptation)

A particle moves back and forth in a horizontal line between two end points. A fixed origin \(O\) is the midpoint between these end points. Its displacement \(x\) metres from \(O\) at time \(t\) seconds is \(x=4\sin(2t+\frac\pi3)\), where \(0\leq t\leq\pi\).

1. Determine
   1. the amplitude;[1]
   2. the initial displacement;[2]
   3. the value of \(t\) when the particle first passes through \(O\).[2]
2. More generally, a particle moves along a straight line with displacement \(s=A\sin(nt+b)\) metres at time \(t\) seconds, where \(t\geq0\), \(A,n>0\) and \(-\pi\leq b\leq\pi\).
   Its velocity and acceleration are \(v=\frac{\mathrm{d}s}{\mathrm{d}t}\) and \(a=\frac{\mathrm{d}v}{\mathrm{d}t}\), respectively.
   Find expressions for \(\frac{\mathrm{d}s}{\mathrm{d}t}\) and \(\frac{\mathrm{d}^2s}{\mathrm{d}t^2}\), and hence show that \(a=-n^2s\).[2]
3. 1. Use the chain rule to show that \(a=v\frac{\mathrm{d}v}{\mathrm{d}s}\), where \(v\) is velocity.[1]
   2. The particle is instantaneously at rest when \(s=A\). By separating the variables in \(v\frac{\mathrm{d}v}{\mathrm{d}s}=-n^2s\), show that \(v^2=n^2(A^2-s^2)\).
   3. Hence, or otherwise, find the particle’s maximum speed.[2]