4 A particle of mass 0.5 kg is initially at point $O$. It moves from rest along the $x$-axis under the influence of two forces $F _ { 1 } \mathrm {~N}$ and $F _ { 2 } \mathrm {~N}$ which act parallel to the $x$-axis. At time $t$ seconds the velocity of the particle is $v \mathrm {~ms} ^ { - 1 }$.\\
$F _ { 1 }$ is acting in the direction of motion of the particle and $F _ { 2 }$ is resisting motion.\\
In an initial model

\begin{itemize}
  \item $F _ { 1 }$ is proportional to $t$ with constant of proportionality $\lambda > 0$,
  \item $F _ { 2 }$ is proportional to $v$ with constant of proportionality $\mu > 0$.
\begin{enumerate}[label=(\alph*)]
\item Show that the motion of the particle can be modelled by the following differential equation.
\end{itemize}

$$\frac { 1 \mathrm {~d} v } { 2 \mathrm {~d} t } = \lambda t - \mu v$$
\item Solve the differential equation in part (a), giving the particular solution for $v$ in terms of $t$, $\lambda$ and $\mu$.

You are now given that $\lambda = 2$ and $\mu = 1$.
\item Find a formula for an approximation for $v$ in terms of $t$ when $t$ is large.

In a refined model

\begin{itemize}
  \item $F _ { 1 }$ is constant, acting in the direction of motion with magnitude 2 N ,
  \item $F _ { 2 }$ is as before with $\mu = 1$.
\item Write down a differential equation for the refined model.
\item Without solving the differential equation in part (d), write down what will happen to the velocity in the long term according to this refined model.
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q4 [13]}}