5 A particle \(P\) moves in a straight line from a fixed point \(O\). The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) at time \(t \mathrm {~s}\) is given by $$v = t ^ { 2 } - 8 t + 12 \quad \text { for } 0 \leqslant t \leqslant 8$$

1. Find the minimum velocity of \(P\).
2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 8\).
   \includegraphics[max width=\textwidth, alt={}, center]{555678d3-f37d-4822-a005-de8c6094dc50-12_401_1102_260_520} Two particles \(A\) and \(B\), of masses 0.4 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is held on a smooth plane inclined at an angle of \(\theta ^ { \circ }\) to the horizontal. The string passes over a small smooth pulley \(P\) fixed at the top of the plane, and \(B\) hangs freely 0.5 m above horizontal ground (see diagram). The particles are released from rest with both sections of the string taut.
3. Given that the system is in equilibrium, find \(\theta\).
4. It is given instead that \(\theta = 20\). In the subsequent motion particle \(A\) does not reach \(P\) and \(B\) remains at rest after reaching the ground.
   (a) Find the tension in the string and the acceleration of the system.
   (b) Find the speed of \(A\) at the instant \(B\) reaches the ground.
   (c) Use an energy method to find the total distance \(A\) moves up the plane before coming to instantaneous rest.
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