2 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 \(v = 5 , R = 2\).
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\item Show that, according to the model, \(\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 10 f - 4 v } { 45 }\).
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- By solving the differential equation in part (a), show that \(v = \frac { 1 } { 2 } \left( 5 f - ( 5 f - 2 u ) \mathrm { e } ^ { - \frac { 4 } { 45 } t } \right)\).
- Describe briefly how, according to the model, the speed of \(P\) varies over time in each of the following cases.
The flat surface of a smooth solid hemisphere of radius \(r\) is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by \(\gamma\). \(O\) is the centre of the flat surface of the hemisphere.
A particle \(P\) is held at a point on the surface of the hemisphere such that the angle between \(O P\) and the upward vertical through \(O\) is \(\alpha\), where \(\cos \alpha = \frac { 3 } { 4 }\).
\(P\) is then released from rest. \(F\) is the point on the plane where \(P\) first hits the plane (see diagram).
- Find an exact expression for the distance \(O F\).
The acceleration due to gravity on and near the surface of the planet Earth is roughly \(6 \gamma\).
- Explain whether \(O F\) would increase, decrease or remain unchanged if the action were repeated on the planet Earth.