Challenging +1.2 This is a multi-stage collision problem requiring systematic application of conservation of momentum and Newton's restitution law across three collision events, then solving a resulting equation. While it involves several steps and careful bookkeeping of velocities through multiple collisions, the techniques are standard for Further Maths mechanics: no novel geometric insight or proof is required, just methodical application of formulae. The algebraic manipulation is moderate, making this above average but not exceptionally challenging for FM students.
5 Two uniform small smooth spheres \(A\) and \(B\), of equal radii, have masses \(2 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane. Sphere \(A\) is projected directly towards \(B\) with speed \(u\). After the collision \(B\) goes on to collide directly with a fixed smooth vertical barrier, before colliding with \(A\) again. The coefficient of restitution between \(A\) and \(B\) is \(\frac { 2 } { 3 }\) and the coefficient of restitution between \(B\) and the barrier is \(e\). After the second collision between \(A\) and \(B\), the speed of \(B\) is five times the speed of \(A\). Find the two possible values of \(e\).
5 Two uniform small smooth spheres $A$ and $B$, of equal radii, have masses $2 m$ and $m$ respectively. They lie at rest on a smooth horizontal plane. Sphere $A$ is projected directly towards $B$ with speed $u$. After the collision $B$ goes on to collide directly with a fixed smooth vertical barrier, before colliding with $A$ again. The coefficient of restitution between $A$ and $B$ is $\frac { 2 } { 3 }$ and the coefficient of restitution between $B$ and the barrier is $e$. After the second collision between $A$ and $B$, the speed of $B$ is five times the speed of $A$. Find the two possible values of $e$.
\hfill \mbox{\textit{CAIE FP2 2013 Q5 [11]}}