5.

\begin{figure}[h]
\begin{center}
  \begin{tikzpicture}[>=latex, line join=round, line cap=round]

    % Define parameters for easy adjustment
    \def\HeightA{3.6}       % Height of point A
    \def\Px{3.4}            % X-coordinate of point P
    \def\Pradius{0.14}      % Radius of the black dot P
    
    % Define main coordinates
    \coordinate (O) at (0, 0);
    \coordinate (A) at (0, \HeightA);
    \coordinate (P) at (\Px, \Pradius); % Center of P is above the line by its radius

    % Draw horizontal line (ground)
    \draw (-4.5, 0) -- (4.2, 0);

    % Draw vertical dashed line
    \draw[densely dashed, semithick] (O) -- (A);

    % Draw the solid slanted line
    \draw[semithick] (A) -- (P) node[midway, above right=1pt] {$a$};

    % Draw the angle arc and label
    % Calculate the angle of the line AP from the vertical
    \pgfmathsetmacro{\myangle}{atan(\Px / (\HeightA - \Pradius))}
    \begin{scope}[shift={(A)}]
        % Draw arc starting from the vertical dashed line (-90 degrees)
        \draw (0, -1.6) arc (-90:{-90 + \myangle}:1.6);
        % Place the angle label roughly bisecting the arc
        \node at ({-90 + \myangle/2 + 2}:2.0) {$\theta^\circ$};
    \end{scope}

    % Labels for A and O
    \node[above=2pt] at (A) {$A$};
    \node[below=2pt] at (O) {$O$};

    % Draw the particle P (resting exactly on the horizontal line) and its label
    \fill (P) circle (\Pradius);
    \node[right=3pt] at (P) {$P$};

\end{tikzpicture}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held at rest vertically below $O$, with the string taut, as shown in Figure 4.

The particle is then projected horizontally with speed $u$, where $u > \sqrt { 2 a g }$\\
Air resistance is modelled as being negligible.\\
At the instant when the string makes an angle $\theta$ with the upward vertical through $O$, the speed of $P$ is $v$ and the string goes slack.
\begin{enumerate}[label=(\alph*)]
\item Show that $3 v ^ { 2 } = u ^ { 2 } - 2 a g$

From the instant when the string goes slack to the instant when $O P$ is next horizontal, $P$ moves as a projectile.

The time from the instant when the string goes slack to the instant when $O P$ is next horizontal is $T$

Given that $\theta = 30 ^ { \circ }$
\item show that $T = \frac { 2 v } { g }$
\item Hence, show that the string goes taut again when it is next horizontal.
\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2022 Q5 [13]}}