5. A toy car of mass 0.5 kg is attached to one end $A$ of a light elastic string $A B$, of natural length 1.5 m and modulus of elasticity 27 N . Initially the car is at rest on a smooth horizontal floor and the string lies in a straight line with $A B = 1.5 \mathrm {~m}$. The end $B$ is moved in a straight horizontal line directly away from the car, with constant speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At time $t$ seconds after $B$ starts to move, the extension of the string is $x$ metres and the car has moved a distance $y$ metres. The effect of air resistance on the car can be ignored.

By modelling the car as a particle, show that, while the string remains taut,
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item $x + y = u t$
\item $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 36 x = 0$
\end{enumerate}\item Hence show that the string becomes slack when $t = \frac { \pi } { 6 }$
\item Find, in terms of $u$, the speed of the car when $t = \frac { \pi } { 12 }$
\item Find, in terms of $u$, the distance the car has travelled when it first reaches end $B$ of the string.\\

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\hfill \mbox{\textit{Edexcel M4 2016 Q5 [17]}}