8 A helicopter travels at a constant height above the sea. It passes directly over a lighthouse with position vector $( 500 \mathbf { i } + 200 \mathbf { j } )$ metres relative to the origin, with a velocity of $( - 17.5 \mathbf { i } - 27 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }$. The helicopter moves with a constant acceleration of $( 0.5 \mathbf { i } + 0.6 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. The unit vectors $\mathbf { i }$ and $\mathbf { j }$ are directed east and north respectively.
\begin{enumerate}[label=(\alph*)]
\item Find the position vector of the helicopter $t$ seconds after it has passed over the lighthouse.
\item The position vector of a rock is $( 200 \mathbf { i } - 400 \mathbf { j } )$ metres relative to the origin. Show that the helicopter passes directly over the rock, and state the time that it takes for the helicopter to move from the lighthouse to the rock.
\item Find the average velocity of the helicopter as it moves from the lighthouse to the rock.
\item Is the magnitude of the average velocity equal to the average speed of the helicopter? Give a reason for your answer.
\end{enumerate}

\hfill \mbox{\textit{AQA M1 2013 Q8 [15]}}