7 A force of magnitude \(0.4 t \mathrm {~N}\), applied at an angle of \(30 ^ { \circ }\) above the horizontal, acts on a particle \(P\), where \(t \mathrm {~s}\) is the time since the force starts to act. \(P\) is at rest on rough horizontal ground when \(t = 0\). The mass of \(P\) is 0.2 kg and the coefficient of friction between \(P\) and the ground is \(\mu\).

1. Given that \(P\) is about to slip when \(t = 2\), find \(\mu\) and the value of \(t\) for the instant when \(P\) loses contact with the ground.
2. While \(P\) is moving on the ground, it has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\). Show that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = 2.165 t - 4.330$$ where the coefficients are correct to 4 significant figures.
3. Calculate the speed of \(P\) when it loses contact with the ground. \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. }