7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2226b09b-a19d-4253-baaa-3c3d602d0d6d-13_520_1027_296_447}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
A small ball is projected from a fixed point \(O\) so as to hit a target \(T\) which is at a horizontal distance \(9 a\) from \(O\) and at a height \(6 a\) above the level of \(O\). The ball is projected with speed \(\sqrt { } ( 27 a g )\) at an angle \(\theta\) to the horizontal, as shown in Figure 4. The ball is modelled as a particle moving freely under gravity.
- Show that \(\tan ^ { 2 } \theta - 6 \tan \theta + 5 = 0\)
The two possible angles of projection are \(\theta _ { 1 }\) and \(\theta _ { 2 }\), where \(\theta _ { 1 } > \theta _ { 2 }\).
- Find \(\tan \theta _ { 1 }\) and \(\tan \theta _ { 2 }\).
The particle is projected at the larger angle \(\theta _ { 1 }\).
- Show that the time of flight from \(O\) to \(T\) is \(\sqrt { } \left( \frac { 78 a } { g } \right)\).
- Find the speed of the particle immediately before it hits \(T\).