7 A projectile is fired with speed \(u\) from a point \(O\) on a plane which is inclined at an angle \(\alpha\) to the horizontal. The projectile is fired at an angle \(\theta\) to the inclined plane and moves in a vertical plane through a line of greatest slope of the inclined plane. The projectile lands at a point \(P\), lower down the inclined plane, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{eed9842d-cd89-4eb7-b5ba-9380971be196-5_415_1098_495_463}

1. Find, in terms of \(u , g , \theta\) and \(\alpha\), the greatest perpendicular distance of the projectile from the plane.
2. 1. Find, in terms of \(u , g , \theta\) and \(\alpha\), the time of flight from \(O\) to \(P\).
   2. By using the identity \(\cos A \cos B + \sin A \sin B = \cos ( A - B )\), show that the distance \(O P\) is given by \(\frac { 2 u ^ { 2 } \sin \theta \cos ( \theta - \alpha ) } { g \cos ^ { 2 } \alpha }\).
   3. Hence, by using the identity \(2 \sin A \cos B = \sin ( A + B ) + \sin ( A - B )\) or otherwise, show that, as \(\theta\) varies, the maximum possible distance \(O P\) is \(\frac { u ^ { 2 } } { g ( 1 - \sin \alpha ) }\).
      (5 marks)