9 At a theme park, a light elastic rope is used to bring a carriage to rest at the end of a ride.

The carriage has mass 200 kg and is travelling at $8 \mathrm {~ms} ^ { - 1 }$ when the elastic rope is attached to the carriage as it passes over a point $O$. The other end of the elastic rope is fixed to the point $O$. The carriage then moves along a horizontal surface until it is brought to rest. The elastic rope is then detached so that the carriage remains at rest.

The elastic rope has natural length 6 m and modulus of elasticity 1800 N . The rope, once taut, remains horizontal throughout the motion.
\begin{enumerate}[label=(\alph*)]
\item Calculate the elastic potential energy of the rope when the carriage is 10 m from $O$.\\
(3 marks)
\item A student's simple model assumes that there are no resistance forces acting on the carriage so that it is brought to rest by the elastic rope alone.

Find the distance of the carriage from $O$ when it is brought to rest.
\item The student improves the model by also including a constant resistance force of 800 N which acts while the carriage is in motion.

Find the distance of the carriage from $O$ when it is brought to rest.\\
(8 marks)
\end{enumerate}

\hfill \mbox{\textit{AQA M2 2011 Q9 [14]}}