Challenging +1.2 This is a standard Further Maths mechanics problem involving successive collisions with coefficient of restitution. While it requires multiple steps (conservation of momentum, Newton's law of restitution, kinetic energy condition, and checking for further collisions), the techniques are routine for FM students and the problem structure is typical of exam questions. The 'show that' part provides scaffolding, and the logic for deducing no further collisions is straightforward velocity comparison. More challenging than average A-level but standard for Further Maths collision problems.
2 Three uniform small smooth spheres, \(A , B\) and \(C\), have equal radii. Their masses are \(4 m , 2 m\) and \(m\) respectively. They lie in a straight line on a smooth horizontal surface with \(B\) between \(A\) and \(C\). Initially \(A\) is moving towards \(B\) with speed \(u , B\) is at rest and \(C\) is moving in the same direction as \(A\) with speed \(\frac { 1 } { 2 } u\). The coefficient of restitution between any two of the spheres is \(e\). The first collision is between \(A\) and \(B\). In this collision sphere \(A\) loses three-quarters of its kinetic energy. Show that \(e = \frac { 1 } { 2 }\).
Find the speed of \(B\) after its collision with \(C\) and deduce that there are no further collisions between the spheres.
2 Three uniform small smooth spheres, $A , B$ and $C$, have equal radii. Their masses are $4 m , 2 m$ and $m$ respectively. They lie in a straight line on a smooth horizontal surface with $B$ between $A$ and $C$. Initially $A$ is moving towards $B$ with speed $u , B$ is at rest and $C$ is moving in the same direction as $A$ with speed $\frac { 1 } { 2 } u$. The coefficient of restitution between any two of the spheres is $e$. The first collision is between $A$ and $B$. In this collision sphere $A$ loses three-quarters of its kinetic energy. Show that $e = \frac { 1 } { 2 }$.
Find the speed of $B$ after its collision with $C$ and deduce that there are no further collisions between the spheres.
\hfill \mbox{\textit{CAIE FP2 2013 Q2 [11]}}