6 A particle, \(P\), travels in a straight line, starting from a point \(O\) with velocity \(6 \mathrm {~ms} ^ { - 1 }\). The acceleration of \(P\) at time \(t \mathrm {~s}\) after leaving \(O\) is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where $$\begin{array} { l l } a = - 1.5 t ^ { \frac { 1 } { 2 } } & \text { for } 0 \leqslant t \leqslant 1 ,
a = 1.5 t ^ { \frac { 1 } { 2 } } - 3 t ^ { - \frac { 1 } { 2 } } & \text { for } t > 1 . \end{array}$$

1. Find the velocity of \(P\) at \(t = 1\).
2. Given that there is no change in the velocity of \(P\) when \(t = 1\), find an expression for the velocity of \(P\) for \(t > 1\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-11_2725_35_99_20}
3. Given that the velocity of \(P\) is positive for \(t \leqslant 4\), find the total distance travelled between \(t = 0\) and \(t = 4\).
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_723_762_248_653} Two particles, \(A\) and \(B\), of masses 0.2 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a small fixed smooth pulley which is attached to the bottom of a rough plane inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = 0.6\). Particle \(A\) lies on the plane, and particle \(B\) hangs vertically below the pulley, 0.25 m above horizontal ground. The string between \(A\) and the pulley is parallel to a line of greatest slope of the plane (see diagram). The coefficient of friction between \(A\) and the plane is 1.125 . Particle \(A\) is released from rest.
4. Find the tension in the string and the magnitude of the acceleration of the particles.
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-12_2716_38_109_2012}
5. When \(B\) reaches the ground, it comes to rest. Find the total distance that \(A\) travels down the plane from when it is released until it comes to rest. You may assume that \(A\) does not reach the pulley.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.
   \includegraphics[max width=\textwidth, alt={}, center]{145d93bd-7f56-4e8c-a646-938330511347-14_2715_31_106_2016}