Challenging +1.2 This is a standard two-collision momentum problem requiring systematic application of conservation of momentum and Newton's restitution law twice. While it involves algebraic manipulation with a parameter λ and requires careful bookkeeping across two collisions, the techniques are routine for Further Maths students and the problem structure is predictable. The 'show that' part guides students through the first collision, making the second part more accessible.
2 Three uniform small smooth spheres \(A , B\) and \(C\), of equal radii and of masses \(4 m , \lambda m\) and \(m\) respectively, are at rest in a straight line on a smooth horizontal plane, with \(B\) between \(A\) and \(C\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). The coefficient of restitution between \(A\) and \(B\), and between \(B\) and \(C\), is \(\frac { 1 } { 2 }\). Show that the speed of \(B\) after it is struck by \(A\) is \(\frac { 6 u } { \lambda + 4 }\).
Given that the speed of \(C\) after it is struck by \(B\) is \(u\), find the value of \(\lambda\).
2 Three uniform small smooth spheres $A , B$ and $C$, of equal radii and of masses $4 m , \lambda m$ and $m$ respectively, are at rest in a straight line on a smooth horizontal plane, with $B$ between $A$ and $C$. Sphere $A$ is projected directly towards $B$ with speed $u$. The coefficient of restitution between $A$ and $B$, and between $B$ and $C$, is $\frac { 1 } { 2 }$. Show that the speed of $B$ after it is struck by $A$ is $\frac { 6 u } { \lambda + 4 }$.
Given that the speed of $C$ after it is struck by $B$ is $u$, find the value of $\lambda$.
\hfill \mbox{\textit{CAIE FP2 2013 Q2 [9]}}