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\includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-7_127_1147_260_459} A particle \(P\) is moving along a straight line with constant acceleration. Initially the particle is at \(O\). After 9 s , \(P\) is at a point \(A\), where \(O A = 18 \mathrm {~m}\) (see diagram) and the velocity of \(P\) at \(A\) is \(8 \mathrm {~ms} ^ { - 1 }\) in the direction \(\overrightarrow { O A }\).

1. (a) Show that the initial speed of \(P\) is \(4 \mathrm {~ms} ^ { - 1 }\).
   (b) Find the acceleration of \(P\).
   \(B\) is a point on the line such that \(O B = 10 \mathrm {~m}\), as shown in the diagram.
2. Show that \(P\) is never at point \(B\). A second particle \(Q\) moves along the same straight line, but has variable acceleration. Initially \(Q\) is at \(O\), and the displacement of \(Q\) from \(O\) at time \(t\) seconds is given by $$x = a t ^ { 3 } + b t ^ { 2 } + c t$$ where \(a\), \(b\) and \(c\) are constants.
   It is given that
   • the velocity and acceleration of \(Q\) at the point \(O\) are the same as those of \(P\) at \(O\),
   • \(\quad Q\) reaches the point \(A\) when \(t = 6\).
   • Find the velocity of \(Q\) at \(A\).
   \section*{OCR} \section*{Oxford Cambridge and RSA}