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\includegraphics[max width=\textwidth, alt={}, center]{022776e2-80bb-4e73-a309-8cd972ae7377-4_341_572_258_790} A particle \(P\) of mass 0.2 kg is projected with velocity \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) upwards along a line of greatest slope on a plane inclined at \(30 ^ { \circ }\) to the horizontal (see diagram). Air resistance of magnitude \(0.5 v \mathrm {~N}\) opposes the motion of \(P\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\) after projection. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 \sqrt { } 3 }\). The particle \(P\) reaches a position of instantaneous rest when \(t = T\).

1. Show that, while \(P\) is moving up the plane, \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - 2.5 ( 3 + v )\).
2. Calculate \(T\).
3. Calculate the speed of \(P\) when \(t = 2 T\). \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. }