A particle $P$ moves in a straight line so that, at time $t$ seconds, its acceleration $a$ m s$^{-2}$ is given by

$$a = \begin{cases}
4t - t^2, & 0 \leq t \leq 3, \\
\frac{27}{t^2}, & t > 3.
\end{cases}$$

At $t = 0$, $P$ is at rest. Find the speed of $P$ when

\begin{enumerate}[label=(\alph*)]
\item $t = 3$,
[3]
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item $t = 6$.
[5]
\end{enumerate}

Figure 1 shows the path taken by a cyclist in travelling on a section of a road. When the cyclist comes to the point $A$ on the top of a hill, she is travelling at 8 m s$^{-1}$. She descends a vertical distance of 20 m to the bottom of the hill. The road then rises to the point $B$ through a vertical distance of 12 m. When she reaches the point $B$, her speed is 5 m s$^{-1}$. The total mass of the cyclist and the cycle is 80 kg and the total distance along the road from $A$ to $B$ is 500 m. By modelling the resistance to the motion of the cyclist as of constant magnitude 20 N,

\begin{enumerate}[label=(\alph*)]
\item find the work done by the cyclist in moving from $A$ to $B$.
[5]
\end{enumerate}

At $B$ the road is horizontal. Given that at $B$ the cyclist is accelerating at 0.5 m s$^{-2}$,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the power generated by the cyclist at $B$.
[4]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M2  Q2 [17]}}