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\includegraphics[max width=\textwidth, alt={}, center]{a8c1e5b3-4d8b-4795-9e9f-4c0db374112e-5_447_693_255_726} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length 0.7 m . A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(O P\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\).
2. Given that the speed of \(Q\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(O Q\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v ^ { 2 }\) in terms of \(\theta\), and show that the tension in the string \(O Q\) is \(14.7 m ( 1 + 2 \cos \theta ) \mathrm { N }\), where \(m \mathrm {~kg}\) is the mass of \(Q\).
3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(O Q\) becomes slack.
4. Show that \(V ^ { 2 } = 0.8575\), where \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(Q\) when it reaches its greatest height (after the string \(O Q\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position.