4 A smooth sphere, with centre \(O\) and radius \(a\), has its lowest point fixed on a horizontal plane. A particle \(P\) of mass \(m\) is projected horizontally with speed \(u\) from the highest point on the outer surface of the sphere. In the subsequent motion, \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\). Show that, while \(P\) remains in contact with the sphere, the magnitude of the reaction of the sphere on
\(P\) is \(m g ( 3 \cos \theta - 2 ) - \frac { m u ^ { 2 } } { a }\). The particle loses contact with the surface of the sphere when \(\theta = \alpha\). Given that \(u = \frac { 1 } { 2 } \sqrt { } ( g a )\), find

1. \(\cos \alpha\),
2. the vertical component of the velocity of \(P\) as it strikes the horizontal plane.