12 A girl is practising netball.\\
She throws the ball from a height of 1.5 m above horizontal ground and aims to get the ball through a hoop.\\
The hoop is 2.5 m vertically above the ground and is 6 m horizontally from the point of projection.

The situation is modelled as follows.

\begin{itemize}
  \item The initial velocity of the ball has magnitude $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
  \item The angle of projection is $40 ^ { \circ }$.
  \item The ball is modelled as a particle.
  \item The hoop is modelled as a point.
\end{itemize}

This is shown on the diagram below.\\
\includegraphics[max width=\textwidth, alt={}, center]{ec83c2c5-f8f8-4357-abfa-d40bc1d026b4-09_375_1207_1119_278}
\begin{enumerate}[label=(\alph*)]
\item For $U = 10$, find
\begin{enumerate}[label=(\roman*)]
\item the greatest height above the ground reached by the ball
\item the distance between the ball and the hoop when the ball is vertically above the hoop.
\end{enumerate}\item Calculate the value of $U$ which allows her to hit the hoop.
\item How appropriate is this model for predicting the path of the ball when it is thrown by the girl?
\item Suggest one improvement that might be made to this model.
\end{enumerate}

\hfill \mbox{\textit{OCR H240/03  Q12 [14]}}