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\includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-10_474_853_264_242} A particle \(P\) moves freely under gravity in the plane of a fixed horizontal axis \(O x\), which lies on horizontal ground, and a fixed vertical axis \(O y . P\) is projected from \(O\) with a velocity whose components along \(O x\) and \(O y\) are \(U\) and \(V\), respectively. \(P\) returns to the ground at a point \(C\).

1. Determine, in terms of \(U , V\) and \(g\), the distance \(O C\).
   \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-10_478_851_1151_244}
   \(P\) passes through two points \(A\) and \(B\), each at a height \(h\) above the ground and a distance \(d\) apart, as shown in the diagram.
2. Write down the horizontal and vertical components of the velocity of \(P\) at \(A\).
3. Hence determine an expression for \(d\) in terms of \(U , V , g\) and \(h\).
4. Given that the direction of motion of \(P\) as it passes through \(A\) is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 1 } { 2 }\), determine an expression for \(V\) in terms of \(g , d\) and \(h\).