6 Particles \(A\) of mass \(2 m\) and \(B\) of mass \(m\) are on a smooth horizontal floor. \(A\) is moving with speed \(u\) directly towards a vertical wall, and \(B\) is at rest between \(A\) and the wall (see diagram).
\includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-5_224_828_354_244} A collides directly with \(B\). The coefficient of restitution in this collision is \(\frac { 1 } { 2 }\).
\(B\) then collides with the wall, rebounds, and collides with \(A\) for a second time.

1. Show that the speed of \(B\) after its second collision with \(A\) is \(\frac { 1 } { 2 } u\). The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall. The second collision between \(A\) and \(B\) occurs at a distance \(\frac { 1 } { 5 } d\) from the wall.
2. Find the coefficient of restitution for the collision between \(B\) and the wall.