Challenging +1.8 This is a challenging two-part mechanics problem requiring sequential collision analysis with both momentum conservation and coefficient of restitution equations, plus an energy loss condition. While the techniques are standard A-level Further Maths material, the problem demands careful algebraic manipulation across multiple collisions and the energy condition adds complexity beyond routine exercises. The second part requires solving simultaneous equations with the restitution formula, which is non-trivial but systematic.
3 Three small smooth spheres \(A , B\) and \(C\) have equal radii and have masses \(m , 9 m\) and \(k m\) respectively. They are at rest on a smooth horizontal table and lie in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between any pair of the spheres is \(e\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Given that half of the total kinetic energy is lost as result of the collision between \(A\) and \(B\), find the value of \(e\).
After \(B\) and \(C\) collide they move in the same direction and the speed of \(C\) is twice the speed of \(B\). Find the value of \(k\).
3 Three small smooth spheres $A , B$ and $C$ have equal radii and have masses $m , 9 m$ and $k m$ respectively. They are at rest on a smooth horizontal table and lie in a straight line with $B$ between $A$ and $C$. The coefficient of restitution between any pair of the spheres is $e$. Sphere $A$ is projected directly towards $B$ with speed $u$. Given that half of the total kinetic energy is lost as result of the collision between $A$ and $B$, find the value of $e$.
After $B$ and $C$ collide they move in the same direction and the speed of $C$ is twice the speed of $B$. Find the value of $k$.
\hfill \mbox{\textit{CAIE FP2 2014 Q3 [10]}}