3 A particle of mass 2 kg moves along the $x$-axis. At time $t$ seconds the velocity of the particle is $v \mathrm {~ms} ^ { - 1 }$.

The particle is subject to two forces.

\begin{itemize}
  \item One acts in the positive $x$-direction with magnitude $\frac { 1 } { 2 } t \mathrm {~N}$.
  \item One acts in the negative $x$-direction with magnitude $v \mathrm {~N}$.
\begin{enumerate}[label=(\alph*)]
\item Show that the motion of the particle can be modelled by the differential equation
\end{itemize}

$$\frac { \mathrm { d } v } { \mathrm {~d} t } + \frac { 1 } { 2 } v = \frac { 1 } { 4 } t$$

The particle is at rest when $t = 0$.
\item Find $v$ in terms of $t$.
\item Find the velocity of the particle when $t = 2$.

When $t = 2$ the force acting in the positive $x$-direction is replaced by a constant force of magnitude $\frac { 1 } { 2 } \mathrm {~N}$ in the same direction.
\item Refine the differential equation given in part (a) to model the motion for $t \geqslant 2$.
\item Use the refined model from part (d) to find an exact expression for $v$ in terms of $t$ for $t \geqslant 2$.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q3 [11]}}