5 A particle \(P\) of mass 4.5 kg is free to move along the \(x\)-axis. In a model of the motion it is assumed that \(P\) is acted on by two forces:

• a constant force of magnitude \(f \mathrm {~N}\) in the positive \(x\) direction;
• a resistance to motion, \(R \mathrm {~N}\), whose magnitude is proportional to the speed of \(P\).

At time \(t\) seconds the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). When \(t = 0 , P\) is at the origin \(O\) and is moving in the positive direction with speed \(u \mathrm {~ms} ^ { - 1 }\), and when \(v = 5 , R = 2\).

1. Show that, according to the model, \(\frac { d v } { d t } = \frac { 10 f - 4 v } { 45 }\).
2. 1. By solving the differential equation in part (a), show that \(\mathrm { v } = \frac { 1 } { 2 } \left( 5 \mathrm { f } - ( 5 \mathrm { f } - 2 \mathrm { u } ) \mathrm { e } ^ { - \frac { 4 } { 45 } \mathrm { t } } \right)\).
   2. Describe briefly how, according to the model, the speed of \(P\) varies over time in each of the following cases.
      • \(\mathrm { u } < 2.5 \mathrm { f }\)
3. \(\mathrm { u } = 2.5 \mathrm { f }\)
4. \(u > 2.5 f\)
5. In the case where \(\mathrm { u } = 2 \mathrm { f }\), find in terms of \(f\) the exact displacement of \(P\) from \(O\) when \(t = 9\).