M3 June 2007 Q3
3. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the earth. The earth is modelled as a fixed sphere of radius \(R\). When \(S\) is at a distance \(x\) from the centre of the earth, the force exerted by the earth on \(S\) is directed towards the centre of the earth and has magnitude \(\dfrac{k}{x^2}\), where \(k\) is a constant.
(a) Show that \(k = mgR^2\). (2)
Given that \(S\) starts from rest when its distance from the centre of the earth is \(2R\), and that air resistance can be ignored,
(b) find the speed of \(S\) as it crashes into the surface of the earth. (7)
| Scheme | Marks |
|---|---|
| At surface \(\dfrac{k}{R^2} = mg \Rightarrow k = mgR^2\) * cso | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| N2L \(m\ddot{x} = -\dfrac{mgR^2}{x^2}\) | |
| \(v\dfrac{\mathrm{d}v}{\mathrm{d}x} = -\dfrac{gR^2}{x^2}\) or \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1}{2}v^2\right) = -\dfrac{gR^2}{x^2}\) | M1 |
| \(\displaystyle\int v\,\mathrm{d}v = -gR^2\int \dfrac{1}{x^2}\,\mathrm{d}x\) or \(\dfrac{1}{2}v^2 = -gR^2\displaystyle\int \dfrac{1}{x^2}\,\mathrm{d}x\) | M1 |
| \(\dfrac{1}{2}v^2 = \dfrac{gR^2}{x}\ \ (+C)\) | A1 |
| \(x = 2R,\ v = 0 \Rightarrow C = -\dfrac{gR}{2}\) | M1 A1 |
| \(v^2 = \dfrac{2gR^2}{x} - gR\) | |
| At \(x = R\), \(v^2 = \dfrac{2gR^2}{R} - gR\) | M1 |
| \(v = \sqrt{(gR)}\) | A1 |
| (7) | |
| (9 marks) |