Zadanie 2a, Paper 3, maj 2026, TZ3, Matematyka AA IB HL

A scanning head moves on a straight guide, with displacement measured from its central position O. One test run has $x(t)=5\cos(3t-\pi/4)$ metres for 0 ≤ t ≤ 2π/3 seconds. The general model is $s(t)=A\cos(nt+\beta)$, where t≥0, A,n>0 and −π≤β≤π. Write v=ds/dt and a=d²s/dt². Restoring acceleration is modelled by a=−n²s. For an observation taken at a uniformly random phase, the displacement random variable S has density $f_S(s)=1/[\pi\sqrt{A^2-s^2}]$ for −A<s<A and zero otherwise. This density concerns random observation times, not a change to the deterministic motion law.