8 A car is travelling on a straight horizontal road. The velocity of the car, \(v \mathrm {~ms} ^ { - 1 }\), at time \(t\) seconds as it travels past three points, \(P , Q\) and \(R\), is modelled by the equation
\(v = a t ^ { 2 } + b t + c\),
where \(a , b\) and \(c\) are constants.
The car passes \(P\) at time \(t = 0\) with velocity \(8 \mathrm {~ms} ^ { - 1 }\).

1. State the value of \(c\). The car passes \(Q\) at time \(t = 5\) and at that instant its deceleration is \(0.12 \mathrm {~ms} ^ { - 2 }\). The car passes \(R\) at time \(t = 18\) with velocity \(2.96 \mathrm {~ms} ^ { - 1 }\).
2. Determine the values of \(a\) and \(b\).
3. Find, to the nearest metre, the distance between points \(P\) and \(R\).
   \includegraphics[max width=\textwidth, alt={}, center]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-08_469_798_251_244} One end of a light inextensible string is attached to a particle \(A\) of mass 2 kg . The other end of the string is attached to a second particle \(B\) of mass 2.5 kg . Particle \(A\) is in contact with a rough plane inclined at \(\theta\) to the horizontal, where \(\cos \theta = \frac { 4 } { 5 }\). The string is taut and passes over a small smooth pulley \(P\) at the top of the plane. The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Particle \(B\) hangs freely below \(P\) at a distance 1.5 m above horizontal ground, as shown in the diagram. The coefficient of friction between \(A\) and the plane is \(\mu\). The system is released from rest and in the subsequent motion \(B\) hits the ground before \(A\) reaches \(P\). The speed of \(B\) at the instant that it hits the ground is \(1.2 \mathrm {~ms} ^ { - 1 }\).