7. Particle \(A\) has mass \(m\) and particle \(B\) has mass \(2 m\). The particles are moving in the same direction along the same straight line on a smooth horizontal surface.
Particle \(A\) collides directly with particle \(B\).
Immediately before the collision, the speed of \(A\) is \(3 u\) and the speed of \(B\) is \(u\).
The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. 1. Show that the speed of \(B\) immediately after the collision is \(\frac { 5 + 2 e } { 3 } u\)
   2. Find the speed of \(A\) immediately after the collision. After the collision, \(B\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(B\).
      The coefficient of restitution between \(B\) and the wall is \(\frac { 1 } { 3 }\)
      Particle \(B\) rebounds and there is a second collision between \(A\) and \(B\).
      The first collision between \(A\) and \(B\) occurs at a distance \(d\) from the wall.
      The time between the two collisions is \(T\).
      Given that \(e = \frac { 1 } { 2 }\)
2. find \(T\) in terms of \(d\) and \(u\).