M3 June 2010 Q2
2. A particle \(P\) of mass \(m\) is above the surface of the Earth at distance \(x\) from the centre of the Earth. The Earth exerts a gravitational force on \(P\). The magnitude of this force is inversely proportional to \(x^2\).
At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a sphere of radius \(R\).
(a) Prove that the magnitude of the gravitational force on \(P\) is \(\dfrac{mgR^2}{x^2}\). (3)
A particle is fired vertically upwards from the surface of the Earth with initial speed \(3U\). At a height \(R\) above the surface of the Earth the speed of the particle is \(U\).
(b) Find \(U\) in terms of \(g\) and \(R\). (7)
| Scheme | Marks |
|---|---|
| \(F = (-)\dfrac{k}{x^2}\) | M1 |
| \(mg = (-)\dfrac{k}{R^2}\) | M1 |
| \(F = \dfrac{mgR^2}{x^2}\) * | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(m\ddot{x} = -\dfrac{mgR^2}{x^2}\) | M1 |
| \(v\dfrac{\mathrm{d}v}{\mathrm{d}x} = -\dfrac{gR^2}{x^2}\) | M1 |
| \(\dfrac{1}{2}v^2 = \displaystyle\int\left(-\dfrac{gR^2}{x^2}\right)\mathrm{d}x\) | M1 dep on 1st M mark |
| \(\dfrac{1}{2}v^2 = \dfrac{gR^2}{x} \quad (+c)\) | A1 |
| \(x = R,\ \ v = 3U \qquad \dfrac{9U^2}{2} = gR + c\) | M1 dep on 3rd M mark |
| \(\dfrac{1}{2}v^2 = \dfrac{gR^2}{x} + \dfrac{9U^2}{2} - gR\) | |
| \(x = 2R,\ \ v = U \qquad \dfrac{1}{2}U^2 = \dfrac{gR^2}{2R} + \dfrac{9U^2}{2} - gR\) | M1 dep on 3rd M mark |
| \(U^2 = \dfrac{gR}{8}\) | |
| \(U = \sqrt{\dfrac{gR}{8}}\) | A1 |
| (7) | |
| (10 marks) |