Challenging +1.2 This is a standard SHM problem requiring application of the velocity formula v² = ω²(a² - x²) to find amplitude and speed, followed by integration to find time between positions. While it involves multiple steps and careful algebraic manipulation, the techniques are routine for Further Maths students and follow predictable patterns without requiring novel insight.
2 The point \(O\) is on the fixed line \(l\). The point \(A\) on \(l\) is such that \(O A = 3 \mathrm {~m}\). A particle \(P\) oscillates on \(l\) in simple harmonic motion with centre \(O\) and period \(\pi\) seconds. When \(P\) is at \(A\) its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed of \(P\) when it is at the point \(B\) on \(l\), where \(O B = 6 \mathrm {~m}\) and \(B\) is on the same side of \(O\) as \(A\).
Find, correct to 2 decimal places, the time, in seconds, taken for \(P\) to travel directly from \(A\) to \(B\).
2 The point $O$ is on the fixed line $l$. The point $A$ on $l$ is such that $O A = 3 \mathrm {~m}$. A particle $P$ oscillates on $l$ in simple harmonic motion with centre $O$ and period $\pi$ seconds. When $P$ is at $A$ its speed is $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the speed of $P$ when it is at the point $B$ on $l$, where $O B = 6 \mathrm {~m}$ and $B$ is on the same side of $O$ as $A$.
Find, correct to 2 decimal places, the time, in seconds, taken for $P$ to travel directly from $A$ to $B$.
\hfill \mbox{\textit{CAIE FP2 2013 Q2 [8]}}