5 A particle \(P\) moves on the \(x\)-axis from the origin \(O\) with an initial velocity of \(- 20 \mathrm {~ms} ^ { - 1 }\). The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t \mathrm {~s}\) after leaving \(O\) is given by \(a = 12 - 2 t\).

1. Sketch a velocity-time graph for \(0 \leqslant t \leqslant 12\), indicating the times when \(P\) is at rest.
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 12\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{167f782c-3047-41f9-90a8-32ccdc19216d-10_410_723_260_717} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
   \end{figure} Fig. 6.1 shows particles \(A\) and \(B\), of masses 4 kg and 3 kg respectively, attached to the ends of a light inextensible string that passes over a small smooth pulley. The pulley is fixed at the top of a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. \(A\) hangs freely below the pulley and \(B\) is on the inclined plane. The string is taut and the section of the string between \(B\) and the pulley is parallel to a line of greatest slope of the plane.
3. It is given that the plane is rough and the particles are in limiting equilibrium. Find the coefficient of friction between \(B\) and the plane.
4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{167f782c-3047-41f9-90a8-32ccdc19216d-11_412_899_276_589} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} It is given instead that the plane is smooth and the particles are released from rest when the difference in the vertical heights of the particles is 1 m (see Fig. 6.2). Use an energy method to find the speed of the particles at the instant when the particles are at the same horizontal level.
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