12 Two small uniform smooth spheres A and B are of equal radius and have masses \(m\) and \(\lambda m\) respectively. The spheres are on a smooth horizontal surface. Sphere A is moving on the surface with velocity \(u _ { 1 } \mathbf { i } + u _ { 2 } \mathbf { j }\) towards B , which is at rest.
The spheres collide obliquely. When the spheres collide, the line joining their centres is parallel to \(\mathbf { i }\). The coefficient of restitution between A and B is \(e\).

1. 1. Explain why, when the spheres collide, the impulse of A on B is in the direction of \(\mathbf { i }\).
   2. Determine this impulse in terms of \(\lambda , m , e\) and \(u _ { 1 }\). The loss in kinetic energy due to the collision between A and B is \(\frac { 1 } { 8 } \mathrm { mu } _ { 1 } { } ^ { 2 }\).
2. Determine the range of possible values of \(\lambda\).