7. \begin{figure}[h] \end{figure} A smooth uniform sphere \(A\) of mass \(m\) is moving with speed \(U\) on a smooth horizontal plane. The sphere \(A\) collides obliquely with a smooth uniform sphere \(B\) of mass \(3 m\) which is at rest on the plane. The two spheres have the same radius. Immediately before the collision, the direction of motion of \(A\) makes an angle \(\alpha\), where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\), with the line joining the centres of the spheres. Immediately after the collision, the direction of motion of \(A\) is perpendicular to its original direction, as shown in Figure 1. The coefficient of restitution between the spheres is \(e\).

1. Show that the speed of \(B\) immediately after the collision is $$\frac { 1 } { 4 } ( 1 + e ) U \cos \alpha$$
2. Show that \(e > \frac { 1 } { 3 }\)
3. Show that \(0 < \tan \alpha \leqslant \frac { 1 } { \sqrt { 2 } }\)