13 A small ball \(B\) moves in the plane of a fixed horizontal axis \(O x\), which lies on horizontal ground, and a fixed vertically upwards axis \(O y . B\) is projected from \(O\) with a velocity whose components along \(O x\) and \(O y\) are \(U \mathrm {~ms} ^ { - 1 }\) and \(V \mathrm {~ms} ^ { - 1 }\), respectively. The units of \(x\) and \(y\) are metres.
\(B\) is modelled as a particle moving freely under gravity.

1. Show that the path of \(B\) has equation \(2 U ^ { 2 } y = 2 U V x - g x ^ { 2 }\). During its motion, \(B\) just clears a vertical wall of height \(\frac { 1 } { 2 } a \mathrm {~m}\) at a horizontal distance \(a \mathrm {~m}\) from \(O\). \(B\) strikes the ground at a horizontal distance \(3 a \mathrm {~m}\) beyond the wall.
2. Determine the angle of projection of \(B\). Give your answer in degrees correct to \(\mathbf { 3 }\) significant figures.
3. Given that the speed of projection of \(B\) is \(54.6 \mathrm {~ms} ^ { - 1 }\), determine the value of \(a\).
4. Hence find the maximum height of \(B\) above the ground during its motion.
5. State one refinement of the model, other than including air resistance, that would make it more realistic. \section*{END OF QUESTION PAPER}