\includegraphics{figure_5}

Particles $P_1$ and $P_2$ are each moving with simple harmonic motion along the same straight line. $P_1$'s motion has centre $C_1$, period $2\pi$ s and amplitude $3$ m; $P_2$'s motion has centre $C_2$, period $\frac{4}{3}\pi$ s and amplitude $4$ m. The points $C_1$ and $C_2$ are $6.5$ m apart. The displacements of $P_1$ and $P_2$ from their centres of oscillation at time $t$ s are denoted by $x_1$ m and $x_2$ m respectively. The diagram shows the positions of the particles at time $t = 0$, when $x_1 = 3$ and $x_2 = 4$.

\begin{enumerate}[label=(\roman*)]
\item State expressions for $x_1$ and $x_2$ in terms of $t$, which are valid until the particles collide. [3]
\end{enumerate}

The particles collide when $t = 5.99$, correct to $3$ significant figures.

\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the distance travelled by $P_2$ before the collision takes place. [4]
\item Find the velocities of $P_1$ and $P_2$ immediately before the collision, and state whether the particles are travelling in the same direction or in opposite directions. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR M3 2010 Q5 [11]}}