Two particles A and B have masses \(m \text{ kg}\) and \(km \text{ kg}\) respectively, where \(k > 1\). The particles are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley and the particles hang vertically below it. Both particles are at a height of 0.81 m above horizontal ground (see diagram). The system is released from rest and particle B reaches the ground 0.9 s later. The particle A does not reach the pulley in its subsequent motion.

(i) Find the value of \(k\) and show that the tension in the string before B reaches the ground is equal to \(12m \text{ N}\).

At the instant when B reaches the ground, the string breaks.

(ii) Show that the speed of A when it reaches the ground is \(5.97 \text{ m s}^{-1}\), correct to 3 significant figures, and find the time taken, after the string breaks, for A to reach the ground.

(iii) Sketch a velocity-time graph for the motion of particle A from the instant when the system is released until A reaches the ground.