4 A light inextensible taut rope, of length 4 m , is attached at one end \(A\) to the centre of the horizontal ceiling of a gym. The other end of the rope \(B\) is being held by a child of mass 35 kg . Initially the child is held at rest with the rope making an angle of \(60 ^ { \circ }\) to the downward vertical and it may be assumed that the child can be modelled as a particle attached to the end of the rope. The child is released at a height 5 m above the horizontal ground.

1. Show that the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the child when the rope makes an angle \(\theta\) with the downward vertical is given by \(v ^ { 2 } = 4 g ( 2 \cos \theta - 1 )\).
2. At the instant when \(\theta = 0 ^ { \circ }\), the child lets go of the rope and moves freely under the influence of gravity only. Determine the speed and direction of the child at the moment that the child reaches the ground.
3. The child returns to the initial position and is released again from rest. Find the value of \(\theta\) when the tension in the rope is three times greater than the tension in the rope at the instant the child is released.