\begin{enumerate}
  \item \hspace{0pt} [In this question, the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are in a vertical plane, $\mathbf { i }$ being horizontal and $\mathbf { j }$ being vertically upwards.]
\end{enumerate}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c5760fa5-3c7f-4e29-87a2-b3b4145b9361-12_689_1042_360_459}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}

At time $t = 0$, a particle $P$ is projected from the point $A$ which has position vector 10j metres with respect to a fixed origin $O$ at ground level. The ground is horizontal. The velocity of projection of $P$ is $( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$, as shown in Figure 3. The particle moves freely under gravity and reaches the ground after $T$ seconds.\\
(a) For $0 \leqslant t \leqslant T$, show that, with respect to $O$, the position vector, $\mathbf { r }$ metres, of $P$ at time $t$ seconds is given by

$$\mathbf { r } = 3 t \mathbf { i } + \left( 10 + 5 t - 4.9 t ^ { 2 } \right) \mathbf { j }$$

(b) Find the value of $T$.\\
(c) Find the velocity of $P$ at time $t$ seconds $( 0 \leqslant t \leqslant T )$.

When $P$ is at the point $B$, the direction of motion of $P$ is $45 ^ { \circ }$ below the horizontal.\\
(d) Find the time taken for $P$ to move from $A$ to $B$.\\
(e) Find the speed of $P$ as it passes through $B$.

\hfill \mbox{\textit{Edexcel M2 2011 Q6 [12]}}