6
\includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-3_579_1518_258_315} The diagram shows the velocity-time graph for a particle \(P\) which travels on a straight line \(A B\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\). The graph consists of five straight line segments. The particle starts from rest when \(t = 0\) at a point \(X\) on the line between \(A\) and \(B\) and moves towards \(A\). The particle comes to rest at \(A\) when \(t = 2.5\).

1. Given that the distance \(X A\) is 4 m , find the greatest speed reached by \(P\) during this stage of the motion. In the second stage, \(P\) starts from rest at \(A\) when \(t = 2.5\) and moves towards \(B\). The distance \(A B\) is 48 m . The particle takes 12 s to travel from \(A\) to \(B\) and comes to rest at \(B\). For the first 2 s of this stage \(P\) accelerates at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
2. the value of \(V\),
3. the value of \(t\) at which \(P\) starts to decelerate during this stage,
4. the deceleration of \(P\) immediately before it reaches \(B\).
   \(7 \quad\) A particle \(P\) travels in a straight line. It passes through the point \(O\) of the line with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\), where \(t\) is in seconds. \(P\) 's velocity after leaving \(O\) is given by $$\left( 0.002 t ^ { 3 } - 0.12 t ^ { 2 } + 1.8 t + 5 \right) \mathrm { m } \mathrm {~s} ^ { - 1 }$$ The velocity of \(P\) is increasing when \(0 < t < T _ { 1 }\) and when \(t > T _ { 2 }\), and the velocity of \(P\) is decreasing when \(T _ { 1 } < t < T _ { 2 }\).
5. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) and the distance \(O P\) when \(t = T _ { 2 }\).
6. Find the velocity of \(P\) when \(t = T _ { 2 }\) and sketch the velocity-time graph for the motion of \(P\).