5 A ball is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\alpha\) above the horizontal so as to hit a point \(P\) on a wall. The ball travels in a vertical plane through the point of projection. During the motion, the horizontal and upward vertical displacements of the ball from the point of projection are \(x\) metres and \(y\) metres respectively.

1. Show that, during the flight, the equation of the trajectory of the ball is given by $$y = x \tan \alpha - \frac { g x ^ { 2 } } { 2 u ^ { 2 } } \left( 1 + \tan ^ { 2 } \alpha \right)$$
2. The ball is projected from a point 1 metre vertically below and \(R\) metres horizontally from the point \(P\).
   1. By taking \(g = 10 \mathrm {~ms} ^ { - 2 }\), show that \(R\) satisfies the equation $$5 R ^ { 2 } \tan ^ { 2 } \alpha - u ^ { 2 } R \tan \alpha + 5 R ^ { 2 } + u ^ { 2 } = 0$$
   2. Hence, given that \(u\) and \(R\) are constants, show that, for \(\tan \alpha\) to have real values, \(R\) must satisfy the inequality $$R ^ { 2 } \leqslant \frac { u ^ { 2 } \left( u ^ { 2 } - 20 \right) } { 100 }$$
   3. Given that \(R = 5\), determine the minimum possible speed of projection.