8 The diagram shows part of a water park slide, \(A B C\).
The slide is in the shape of two circular arcs, \(A B\) and \(B C\), each of radius \(r\).
The point \(A\) is at a height of \(\frac { r } { 4 }\) above \(B\).
The circular \(\operatorname { arc } B C\) has centre \(O\) and \(B\) is vertically above \(O\).
These points are joined as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-12_590_1173_756_443} A child starts from rest at \(A\), moves along the slide past the point \(B\) and then loses contact with the slide at a point \(D\). The angle between the vertical, \(O B\), and \(O D\) is \(\theta\)
Assume that the slide is smooth. 8

1. Show that the speed \(v\) of the child at \(D\) is given by \(v = \sqrt { \frac { g r } { 2 } ( 5 - 4 \cos \theta ) }\), where \(g\) is the acceleration due to gravity. 8
2. Find \(\theta\), giving your answer to the nearest degree.
   8
3. A refined model takes into account air resistance. Explain how taking air resistance into account would affect your answer to part (b).
   [0pt] [2 marks]
   8
4. In reality the slide is not smooth. It has a surface with the same coefficient of friction between the slide and the child for its entire length. Explain why the frictional force experienced by the child is not constant.
   [0pt] [1 mark]