6 Two particles \(A\) and \(B\), of masses 0.8 kg and 0.2 kg respectively, are connected by a light inextensible string. Particle \(A\) is placed on a horizontal surface. The string passes over a small smooth pulley \(P\) fixed at the edge of the surface, and \(B\) hangs freely. The horizontal section of the string, \(A P\), is of length 2.5 m . The particles are released from rest with both sections of the string taut.

1. Given that the surface is smooth, find the time taken for \(A\) to reach the pulley.
2. Given instead that the surface is rough and the coefficient of friction between \(A\) and the surface is 0.1 , find the speed of \(A\) immediately before it reaches the pulley.
   \(7 \quad\) A particle \(P\) moves in a straight line. The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) is given by $$\begin{array} { l l } v = 5 t ( t - 2 ) & \text { for } 0 \leqslant t \leqslant 4
   v = k & \text { for } 4 \leqslant t \leqslant 14
   v = 68 - 2 t & \text { for } 14 \leqslant t \leqslant 20 \end{array}$$ where \(k\) is a constant.
3. Find \(k\).
4. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 20\).
5. Find the set of values of \(t\) for which the acceleration of \(P\) is positive.
6. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 20\).