5. Above the Earth's surface, the magnitude of the gravitational force on a particle due to the Earth is inversely proportional to the square of the distance of the particle from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and the acceleration due to gravity at the Earth's surface is \(g\). A particle \(P\) of mass \(m\) is at a height \(x\) above the surface of the Earth.

1. Show that the magnitude of the gravitational force acting on \(P\) is $$\frac { m g R ^ { 2 } } { ( R + x ) ^ { 2 } }$$ A rocket is fired vertically upwards from the surface of the Earth. When the rocket is at height \(2 R\) above the surface of the Earth its speed is \(\sqrt { } \left( \frac { g R } { 2 } \right)\). You may assume that air resistance can be ignored and that the engine of the rocket is switched off before the rocket reaches height \(R\). Modelling the rocket as a particle,
2. find the speed of the rocket when it was at height \(R\) above the surface of the Earth.