7 A particle \(P\) moves on a straight line. It starts at a point \(O\) on the line and returns to \(O 100 \mathrm {~s}\) later. The velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\) after leaving \(O\), where $$v = 0.0001 t ^ { 3 } - 0.015 t ^ { 2 } + 0.5 t$$

1. Show that \(P\) is instantaneously at rest when \(t = 0 , t = 50\) and \(t = 100\).
2. Find the values of \(v\) at the times for which the acceleration of \(P\) is zero, and sketch the velocitytime graph for \(P\) 's motion for \(0 \leqslant t \leqslant 100\).
3. Find the greatest distance of \(P\) from \(O\) for \(0 \leqslant t \leqslant 100\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
   To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }