Problem

Two uniform smooth spheres, $A$ and $B$, of equal radii are on a horizontal surface. They have masses of $m$ and $km$ respectively. Sphere $A$ is moving with speed $u$ at an angle $\alpha$ with the line of centres when it collides with sphere $B$ which is stationary. Immediately after the collision, sphere $A$ moves with speed $v$ at an angle $2\alpha$ with the line of centres (see diagram).

It is given that $\tan\alpha=\frac34$.

(a) Find $v$ in terms of $u$.

(b) Find the coefficient of restitution between the spheres in terms of $k$.

(c) Find the range of possible values of $k$.