Challenging +1.2 This is a two-collision momentum problem requiring systematic application of conservation of momentum, Newton's restitution law, and energy considerations across multiple stages. While it involves several steps and careful bookkeeping of velocities through two collisions, the techniques are standard for Further Maths mechanics and the problem structure is clearly signposted. The algebraic manipulation is moderate but straightforward.
3 Two uniform small smooth spheres \(A\) and \(B\), of masses \(m\) and \(2 m\) respectively, and with equal radii, are at rest on a smooth horizontal surface. Sphere \(A\) is projected directly towards \(B\) with speed \(u\), and collides with \(B\). After this collision, sphere \(B\) collides directly with a fixed smooth vertical barrier. The total kinetic energy of the spheres after this second collision is equal to one-ninth of its value before the first collision. Given that the coefficient of restitution between \(B\) and the barrier is 0.5 , find the coefficient of restitution between \(A\) and \(B\).
3 Two uniform small smooth spheres $A$ and $B$, of masses $m$ and $2 m$ respectively, and with equal radii, are at rest on a smooth horizontal surface. Sphere $A$ is projected directly towards $B$ with speed $u$, and collides with $B$. After this collision, sphere $B$ collides directly with a fixed smooth vertical barrier. The total kinetic energy of the spheres after this second collision is equal to one-ninth of its value before the first collision. Given that the coefficient of restitution between $B$ and the barrier is 0.5 , find the coefficient of restitution between $A$ and $B$.
\hfill \mbox{\textit{CAIE FP2 2013 Q3 [9]}}