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\includegraphics[max width=\textwidth, alt={}, center]{0cb05368-9ddf-4564-8428-725c77193a1e-4_933_275_689_934} The diagram shows a light elastic string of natural length 0.6 m and modulus of elasticity 5 N with one end attached to a fixed point \(O\). A particle \(P\) of mass 0.2 kg is attached to the other end of the string. \(P\) is held at the point \(A\), which is 0.9 m vertically above \(O\). The particle is released from rest and travels vertically downwards through \(O\) to the point \(C\), where it starts to move upwards. \(B\) is the point of the line \(A C\) where the string first becomes slack.

1. Find the speed of \(P\) at \(B\).
2. The extension of the string when \(P\) is at \(C\) is \(x \mathrm {~m}\).
   (a) Show that \(x ^ { 2 } - 0.48 x - 0.81 = 0\).
   (b) Hence find the distance \(A C\).