Challenging +1.8 This is a multi-stage mechanics problem requiring: (1) energy conservation to find speed at lowest point, (2) momentum conservation for the collision, (3) circular motion dynamics before and after collision, and (4) solving a quadratic equation. While each individual step uses standard A-level mechanics techniques, the combination of multiple concepts (energy, momentum, circular motion) and the need to relate tensions through the collision makes this significantly harder than routine exercises. It's a typical Further Maths mechanics problem requiring careful systematic work through connected stages.
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\includegraphics[max width=\textwidth, alt={}, center]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-08_323_515_260_813}
A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and horizontal. It is projected downwards with speed \(\sqrt { } ( 12 a g )\). At the lowest point of its motion, \(P\) collides directly with a particle \(Q\) of mass \(k m\) which is at rest (see diagram). In the collision, \(P\) and \(Q\) coalesce. The tension in the string immediately after the collision is half of its value immediately before the collision. Find the possible values of \(k\).
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\includegraphics[max width=\textwidth, alt={}, center]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-08_323_515_260_813}
A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle is held with the string taut and horizontal. It is projected downwards with speed $\sqrt { } ( 12 a g )$. At the lowest point of its motion, $P$ collides directly with a particle $Q$ of mass $k m$ which is at rest (see diagram). In the collision, $P$ and $Q$ coalesce. The tension in the string immediately after the collision is half of its value immediately before the collision. Find the possible values of $k$.\\
\hfill \mbox{\textit{CAIE FP2 2018 Q5 [11]}}