11 A ball \(B\) is projected with speed \(V\) at an angle \(\alpha\) above the horizontal from a point \(O\) on horizontal ground. The greatest height of \(B\) above \(O\) is \(H\) and the horizontal range of \(B\) is \(R\). The ball is modelled as a particle moving freely under gravity.
- Show that
- \(H = \frac { V ^ { 2 } } { 2 g } \sin ^ { 2 } \alpha\),
- \(R = \frac { V ^ { 2 } } { g } \sin 2 \alpha\).
- Hence show that \(16 H ^ { 2 } - 8 R _ { 0 } H + R ^ { 2 } = 0\), where \(R _ { 0 }\) is the maximum range for the given speed of projection.
- Given that \(R _ { 0 } = 200 \mathrm {~m}\) and \(R = 192 \mathrm {~m}\), find
- the two possible values of the greatest height of \(B\),
- the corresponding values of the angle of projection.
- State one limitation of the model that could affect your answers to part (iii).
\section*{OCR}
Oxford Cambridge and RSA