6
\includegraphics[max width=\textwidth, alt={}, center]{6dcce6fe-7a19-4c5f-9361-20e7acda458f-10_339_983_258_541} Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(k m\) respectively. Sphere \(A\) is moving with speed \(u\) on a smooth horizontal surface when it collides with sphere \(B\) which is at rest. Immediately before the collision, \(A\) 's direction of motion makes an angle \(\theta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\).

1. Show that the speed of \(B\) after the collision is \(\frac { 4 \mathrm { u } \cos \theta } { 3 ( 1 + \mathrm { k } ) }\).
   70\% of the total kinetic energy of the spheres is lost as a result of the collision.
2. Given that \(\tan \theta = \frac { 1 } { 3 }\), find the value of \(k\).