A solid smooth sphere, with centre \(O\) and radius \(r\), is fixed to a point \(A\) on a horizontal floor. A particle \(P\) is placed on the surface of the sphere at the point \(B\), where \(B\) is vertically above \(A\). The particle is projected horizontally from \(B\) with speed \(\frac { \sqrt { g r } } { 2 }\) and starts to move on the surface of the sphere. When \(O P\) makes an angle \(\theta\) with the upward vertical and \(P\) remains in contact with the sphere, the speed of \(P\) is \(v\).
Show that \(v ^ { 2 } = \frac { g r } { 4 } ( 9 - 8 \cos \theta )\).
The particle leaves the surface of the sphere when \(\theta = \alpha\).
Find the value of \(\cos \alpha\).
After leaving the surface of the sphere, \(P\) moves freely under gravity and hits the floor at the point \(C\).
Given that \(r = 0.5 \mathrm {~m}\),
find, to 2 significant figures, the distance \(A C\).