Standard +0.8 This is a multi-stage collision problem requiring systematic application of conservation of momentum and Newton's restitution law across two collisions, with the added complexity of finding constraints on parameters. While the techniques are standard for Further Maths mechanics, the problem requires careful algebraic manipulation across multiple stages and checking inequality conditions for no further collisions, placing it moderately above average difficulty.
1 Three small spheres, \(A , B\) and \(C\), of masses \(m , k m\) and \(6 m\) respectively, have the same radius. They are at rest on a smooth horizontal surface, in a straight line with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(\frac { 1 } { 2 }\) and the coefficient of restitution between \(B\) and \(C\) is \(e\). Sphere \(A\) is projected towards \(B\) with speed \(u\) and is brought to rest by the subsequent collision. Show that \(k = 2\).
Given that there are no further collisions after \(B\) has collided with \(C\), show that \(e \leqslant \frac { 1 } { 3 }\).
1 Three small spheres, $A , B$ and $C$, of masses $m , k m$ and $6 m$ respectively, have the same radius. They are at rest on a smooth horizontal surface, in a straight line with $B$ between $A$ and $C$. The coefficient of restitution between $A$ and $B$ is $\frac { 1 } { 2 }$ and the coefficient of restitution between $B$ and $C$ is $e$. Sphere $A$ is projected towards $B$ with speed $u$ and is brought to rest by the subsequent collision. Show that $k = 2$.
Given that there are no further collisions after $B$ has collided with $C$, show that $e \leqslant \frac { 1 } { 3 }$.
\hfill \mbox{\textit{CAIE FP2 2011 Q1 [8]}}