4 A smooth uniform sphere \(A\), of mass \(m\), is moving with velocity \(8 u\) in a straight line on a smooth horizontal table. A smooth uniform sphere \(B\), of mass \(4 m\), has the same radius as \(A\) and is moving on the table with velocity \(u\).
\includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-10_200_1148_456_447} The sphere \(A\) collides directly with the sphere \(B\).
The coefficient of restitution between \(A\) and \(B\) is \(e\).

1. 1. Find, in terms of \(u\) and \(e\), the velocities of \(A\) and \(B\) immediately after the collision.
   2. The direction of motion of \(A\) is reversed by the collision. Show that \(e > a\), where \(a\) is a constant to be determined.
2. Subsequently, \(B\) collides with a fixed smooth vertical wall which is at right angles to the direction of motion of \(A\) and \(B\). The coefficient of restitution between \(B\) and the wall is \(\frac { 2 } { 3 }\). The sphere \(B\) collides with \(A\) again after rebounding from the wall.
   Show that \(e < b\), where \(b\) is a constant to be determined.
3. Given that \(e = \frac { 4 } { 7 }\), find, in terms of \(m\) and \(u\), the magnitude of the impulse exerted on \(B\) by the wall.
   [0pt] [3 marks]