6. A small smooth ball \(A\) of mass \(m\) is moving on a horizontal table with speed \(u\) when it collides directly with another small smooth ball \(B\) of mass \(3 m\) which is at rest on the table. The balls have the same radius and the coefficient of restitution between the balls is \(e\). The direction of motion of \(A\) is reversed as a result of the collision.

1. Find, in terms of \(e\) and \(u\). the speeds of \(A\) and \(B\) immediately after the collision. In the subsequent motion \(B\) strikes a vertical wall, which is perpendicular to the direction of motion of \(B\), and rebounds. The coefficient of restitution between \(B\) and the wall is \(\frac { 3 } { 4 }\). Given that there is a second collision between \(A\) and \(B\),
2. find the range of values of \(e\) for which the motion described is possible.
   (6) \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{a89db1ee-073c-43a3-8480-44970e51c6e2-6_345_1133_440_524}
   \end{figure} A straight \(\log A B\) has weight \(W\) and length \(2 a\). A cable is attached to one end \(B\) of the log. The cable lifts the end \(B\) off the ground. The end \(A\) remains in contact with the ground, which is rough and horizontal. The log is in limiting equilibrium. The log makes an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 5 } { 12 }\). The cable makes an angle \(\beta\) to the horizontal, as shown in Fig. 3. The coefficient of friction between the log and the ground is 0.6 . The log is modelled as a uniform rod and the cable as light.
3. Show that the normal reaction on the \(\log\) at \(A\) is \(\frac { 2 } { 5 } W\).
4. Find the value of \(\beta\). The tension in the cable is \(k W\).
5. Find the value of \(k\).