$O$ is a fixed point on a horizontal plane. A particle $P$ of mass $0.25$ kg is released from rest at $O$ and moves in a straight line on the plane. At time $t$ s after release the only horizontal force acting on $P$ has magnitude
$$\frac{1}{2400}(144 - t^2) \text{ N} \quad \text{for } 0 \leqslant t \leqslant 12$$
and
$$\frac{1}{2400}(t^2 - 144) \text{ N} \quad \text{for } t \geqslant 12.$$

The force acts in the direction of $P$'s motion. $P$'s velocity at time $t$ s is $v$ m s$^{-1}$.

\begin{enumerate}[label=(\roman*)]
\item Find an expression for $v$ in terms of $t$, valid for $t \geqslant 12$, and hence show that $v$ is three times greater when $t = 24$ than it is when $t = 12$. [8]
\item Sketch the $(t, v)$ graph for $0 \leqslant t \leqslant 24$. [3]
\end{enumerate}

\hfill \mbox{\textit{OCR M3 2010 Q4 [11]}}