Two particles A and B, of masses \(0.4\text{ kg}\) and \(0.2\text{ kg}\) respectively, are moving down the same line of greatest slope of a smooth plane. The plane is inclined at \(30^\circ\) to the horizontal, and A is higher up the plane than B. When the particles collide, the speeds of A and B are \(3\text{ m s}^{-1}\) and \(2\text{ m s}^{-1}\) respectively. In the collision between the particles, the speed of A is reduced to \(2.5\text{ m s}^{-1}\). Use \(g=10\text{ m s}^{-2}\).

(a) Find the speed of B immediately after the collision. [2]

After the collision, when B has moved \(1.6\text{ m}\) down the plane from the point of collision, it hits a barrier and returns back up the same line of greatest slope. B hits the barrier \(0.4\text{ s}\) after the collision, and when it hits the barrier, its speed is reduced by 90%. The two particles collide again \(0.44\text{ s}\) after their previous collision, and they then coalesce on impact.

(b) Show that the speed of B immediately after it hits the barrier is \(0.5\text{ m s}^{-1}\). Hence find the speed of the combined particle immediately after the second collision between A and B. [7]