3 Three uniform small spheres $A , B$ and $C$ have equal radii and masses $3 m , m$ 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 each pair of spheres is $e$. Sphere $A$ is projected directly towards $B$ with speed $u$.\\
(i) Find, in terms of $u$ and $e$, expressions for the speeds of $A , B$ and $C$ after the first two collisions.\\

(ii) Given that $A$ and $C$ are moving with equal speeds after these two collisions, find the value of $e$. [3]\\

\includegraphics[max width=\textwidth, alt={}, center]{34dd6523-7c0c-4842-bbda-56ad8d3f9766-08_812_520_260_808}

An object consists of two hollow spheres which touch each other, together with a thin uniform $\operatorname { rod } A B$. The rod passes through small holes in the surfaces of the spheres. The rod is fixed to the spheres so that it passes through the centre of the smaller sphere. The end $B$ of the rod is at the centre of the larger sphere. The larger sphere has radius $2 a$ and mass $M$, the smaller sphere has radius $a$ and mass $k M$, and the rod has length $7 a$ and mass $5 M$. A fixed horizontal axis $L$ passes through $A$ and is perpendicular to $A B$ (see diagram).\\

\hfill \mbox{\textit{CAIE FP2 2019 Q3 [9]}}