7 An engineer is modelling the motion of a particle $P$ of mass 0.5 kg in a wind tunnel.\\
$P$ is modelled as travelling in a straight line. The point $O$ is a fixed point within the wind tunnel. The displacement of $P$ from $O$ at time $t$ seconds is $x$ metres, for $t \geqslant 0$.

You are given that $x \geqslant 0$ for all $t \geqslant 0$ and that $P$ does not reach the end of the wind tunnel.\\
If $t \geqslant 0$, then $P$ is subject to three forces which are modelled in the following way.

\begin{itemize}
  \item The first force has a magnitude of $5 ( t + 1 ) \cosh t \mathrm {~N}$ and acts in the positive $x$-direction.
  \item The second force has a magnitude of $0.5 x \mathrm {~N}$ and acts towards $O$.
  \item The third force has a magnitude of $\left| \frac { d x } { d t } \right| \mathrm { N }$ and acts in the direction of motion of the particle.
\begin{enumerate}[label=(\alph*)]
\item The engineer applies the equation " $F = m a$ " to the model of the motion of $P$ and derives the following differential equation.\\
$5 ( t + 1 ) \operatorname { cosht } - 0.5 x + \frac { d x } { d t } = 0.5 \frac { d ^ { 2 } x } { d t ^ { 2 } }$
\begin{enumerate}[label=(\roman*)]
\item Explain the sign of the $\frac { \mathrm { dx } } { \mathrm { dt } }$ term in the engineer's differential equation.
\end{itemize}

When $t = 0$ the displacement of $P$ is 6 m , and it is travelling towards $O$ with a speed of $5 \mathrm {~ms} ^ { - 1 }$.
\item Without attempting to solve the differential equation, find the acceleration of $P$ when $t = 0$.

Let the particular solution to the differential equation in part (a) be a function f such that $\mathrm { x } = \mathrm { f } ( \mathrm { t } )$ for $t \geqslant 0$.

The particular solution to the differential equation can be expressed as a Maclaurin series.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Show that the Maclaurin series for $\mathrm { f } ( t )$ up to and including the term in $t$ is $6 - 5 t$.
\item Use your answer to part (a)(ii) to show that the term in $t ^ { 2 }$ in the Maclaurin series for $\mathrm { f } ( t )$ is $- 3 t ^ { 2 }$.
\item By differentiating the differential equation in part (a) with respect to $t$, show that the term in $t ^ { 3 }$ in the Maclaurin series for $\mathrm { f } ( t )$ is $0.5 t ^ { 3 }$.

You are given that the complete Maclaurin series for the function f is valid for all values of $t \geqslant 0$.\\
After 0.25 seconds $P$ has travelled 1.43 m towards the origin.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item By using the Maclaurin series for $\mathrm { f } ( t )$ up to and including the term in $t ^ { 3 }$, evaluate the suitability of the model for determining the displacement of $P$ from $O$ when $t = 0.25$.
\item Explain why it might not be sensible to use the Maclaurin series for $\mathrm { f } ( t )$ up to and including the term in $t ^ { 3 }$ to evaluate the suitability of the model for determining the displacement of $P$ from $O$ when $t = 10$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 1 2023 Q7 [11]}}