7 A particle \(P\) of mass 0.5 kg moves on a horizontal surface along the straight line \(O A\), in the direction from \(O\) to \(A\). The coefficient of friction between \(P\) and the surface is 0.08 . Air resistance of magnitude \(0.2 v \mathrm {~N}\) opposes the motion, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(P\) at time \(t \mathrm {~s}\). The particle passes through \(O\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when \(t = 0\).

1. Show that \(2.5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = - ( v + 2 )\) and hence find the value of \(t\) when \(v = 0\).
2. Show that \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 6 \mathrm { e } ^ { - 0.4 t } - 2\), where \(x \mathrm {~m}\) is the displacement of \(P\) from \(O\) at time \(t \mathrm {~s}\), and hence find the distance \(O P\) when \(v = 0\). \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. University of 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. }