2 A lunar mapping satellite of mass \(m _ { 1 }\) measured in kg is in an elliptic orbit around the moon, which has mass \(m _ { 2 }\) measured in kg . The effective potential, \(E\), of the satellite is given by $$E = \frac { K ^ { 2 } } { 2 m _ { 1 } r ^ { 2 } } - \frac { G m _ { 1 } m _ { 2 } } { r }$$ where \(r\) measured in metres is the distance of the satellite from the moon, \(G \mathrm { Nm } ^ { 2 } \mathrm {~kg} ^ { - 2 }\) is the universal gravitational constant, and \(K\) is the angular momentum of the satellite. By using dimensional analysis, find the dimensions of:

1. \(E\),
2. \(\quad K\).
   [0pt] [3 marks] \(3 \quad\) A ball is projected from a point \(O\) on horizontal ground with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(30 ^ { \circ }\) above the horizontal. The ball travels in a vertical plane through the point \(O\) and hits a point \(Q\) on a plane which is inclined at \(45 ^ { \circ }\) to the horizontal. The point \(O\) is 6 metres from \(P\), the foot of the inclined plane, as shown in the diagram. The points \(O , P\) and \(Q\) lie in the same vertical plane. The line \(P Q\) is a line of greatest slope of the inclined plane.
   \includegraphics[max width=\textwidth, alt={}, center]{d8c723df-d10a-4fdf-b5ca-ea12633f999a-06_406_1050_568_488}
3. During its flight, the horizontal and upward vertical distances of the ball from \(O\) are \(x\) metres and \(y\) metres respectively. Show that \(x\) and \(y\) satisfy the equation $$y = x \frac { \sqrt { 3 } } { 3 } - \frac { x ^ { 2 } } { 30 }$$ Use \(\cos 30 ^ { \circ } = \frac { \sqrt { 3 } } { 2 }\) and \(\tan 30 ^ { \circ } = \frac { \sqrt { 3 } } { 3 }\).
4. Find the distance \(P Q\).