Challenging +1.2 This is a standard multi-step mechanics problem requiring conservation of momentum, Newton's law of restitution for two collisions, and kinematic calculations. While it involves several stages (sphere-sphere collision, wall impact, second collision), each step uses routine A-level mechanics techniques with no novel insight required. The perfect elasticity and given coefficient of restitution make calculations straightforward, though careful bookkeeping across multiple collision stages elevates it slightly above average difficulty.
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\includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-2_159_707_1443_721}
Two perfectly elastic small smooth spheres \(A\) and \(B\) have masses \(3 m\) and \(m\) respectively. They lie at rest on a smooth horizontal plane with \(B\) at a distance \(a\) from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and \(B\) is between \(A\) and the barrier (see diagram). Sphere \(A\) is projected towards sphere \(B\) with speed \(u\) and, after the collision between the spheres, \(B\) hits the barrier. The coefficient of restitution between \(B\) and the barrier is \(\frac { 1 } { 2 }\). Find the speeds of \(A\) and \(B\) immediately after they first collide, and the distance from the barrier of the point where they collide for the second time.
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\includegraphics[max width=\textwidth, alt={}, center]{f8dd2aee-4ed5-4588-aa03-5dd56d9e7529-2_159_707_1443_721}
Two perfectly elastic small smooth spheres $A$ and $B$ have masses $3 m$ and $m$ respectively. They lie at rest on a smooth horizontal plane with $B$ at a distance $a$ from a smooth vertical barrier. The line of centres of the spheres is perpendicular to the barrier, and $B$ is between $A$ and the barrier (see diagram). Sphere $A$ is projected towards sphere $B$ with speed $u$ and, after the collision between the spheres, $B$ hits the barrier. The coefficient of restitution between $B$ and the barrier is $\frac { 1 } { 2 }$. Find the speeds of $A$ and $B$ immediately after they first collide, and the distance from the barrier of the point where they collide for the second time.
\hfill \mbox{\textit{CAIE FP2 2010 Q3 [9]}}