Challenging +1.2 This is a standard Further Maths mechanics problem involving successive collisions with two different coefficients of restitution. While it requires careful bookkeeping through multiple collision stages and solving simultaneous equations, the techniques (conservation of momentum, Newton's experimental law) are routine for FP2 students. The algebraic manipulation is moderate, making it slightly above average difficulty but well within expected scope.
5 Three uniform small smooth spheres \(A , B\) and \(C\) have equal radii and masses \(3 m , 2 m\) and \(m\) respectively. The spheres are at rest in a straight line on a smooth horizontal surface, with \(B\) between \(A\) and \(C\). The coefficient of restitution between \(A\) and \(B\) is \(e\) and the coefficient of restitution between \(B\) and \(C\) is \(e ^ { \prime }\). Sphere \(A\) is projected directly towards \(B\) with speed \(u\). Show that, after the collision between \(B\) and \(C\), the speed of \(C\) is \(\frac { 2 } { 5 } u ( 1 + e ) \left( 1 + e ^ { \prime } \right)\) and find the corresponding speed of \(B\).
After this collision between \(B\) and \(C\) it is found that each of the three spheres has the same momentum. Find the values of \(e\) and \(e ^ { \prime }\).
5 Three uniform small smooth spheres $A , B$ and $C$ have equal radii and masses $3 m , 2 m$ and $m$ respectively. The spheres are at rest in a straight line on a smooth horizontal surface, with $B$ between $A$ and $C$. The coefficient of restitution between $A$ and $B$ is $e$ and the coefficient of restitution between $B$ and $C$ is $e ^ { \prime }$. Sphere $A$ is projected directly towards $B$ with speed $u$. Show that, after the collision between $B$ and $C$, the speed of $C$ is $\frac { 2 } { 5 } u ( 1 + e ) \left( 1 + e ^ { \prime } \right)$ and find the corresponding speed of $B$.
After this collision between $B$ and $C$ it is found that each of the three spheres has the same momentum. Find the values of $e$ and $e ^ { \prime }$.
\hfill \mbox{\textit{CAIE FP2 2015 Q5 [12]}}