Challenging +1.2 This is a Further Maths circular motion question requiring students to express position in vector form, differentiate to find velocity and acceleration, then compute and compare magnitudes. While it involves multiple calculus steps and vector manipulation, the approach is methodical: use r = l(cos θ, sin θ), apply chain rule with the given dθ/dt = sin θ, and show |v| = |a| through algebraic simplification. The conceptual demand is moderate for FP2 students familiar with parametric differentiation.
1 A line \(O P\) of fixed length \(l\) rotates in a plane about the fixed point \(O\). At time \(t = 0\), the line is at the position \(O A\). At time \(t\), angle \(A O P = \theta\) radians and \(\frac { \mathrm { d } \theta } { \mathrm { d } t } = \sin \theta\). Show that, for all \(t\), the magnitude of the acceleration of \(P\) is equal to the magnitude of its velocity.
1 A line $O P$ of fixed length $l$ rotates in a plane about the fixed point $O$. At time $t = 0$, the line is at the position $O A$. At time $t$, angle $A O P = \theta$ radians and $\frac { \mathrm { d } \theta } { \mathrm { d } t } = \sin \theta$. Show that, for all $t$, the magnitude of the acceleration of $P$ is equal to the magnitude of its velocity.
\hfill \mbox{\textit{CAIE FP2 2009 Q1 [5]}}