M3 January 2008 Q2
2. A particle \(P\) of mass 0.1 kg moves in a straight line on a smooth horizontal table. When \(P\) is a distance \(x\) metres from a fixed point \(O\) on the line, it experiences a force of magnitude \(\dfrac{16}{5x^2}\) N away from \(O\) in the direction \(OP\). Initially \(P\) is at a point 2 m from \(O\) and is moving towards \(O\) with speed 8 m s\(^{-1}\).
Find the distance of \(P\) from \(O\) when \(P\) first comes to rest. (8)
| Scheme | Marks |
|---|---|
| \(m\) ‘\(a\)’ \(= \pm\dfrac{16}{5x^2}\), with acceleration in any form (e.g. \(\dfrac{\mathrm{d}^2x}{\mathrm{d}t^2},\ v\dfrac{\mathrm{d}v}{\mathrm{d}x},\ \dfrac{\mathrm{d}v}{\mathrm{d}t}\) or | B1 |
| Uses \(a = v\dfrac{\mathrm{d}v}{\mathrm{d}x}\) to obtain \(\mathrm{k}v\dfrac{\mathrm{d}v}{\mathrm{d}x} = \pm\mathrm{k}^{\prime}\dfrac{32}{x^2}\) | M1 |
| Separates variables, \(\mathrm{k}\displaystyle\int v\,\mathrm{d}v = \mathrm{k}^{\prime}\int \dfrac{32}{x^2}\,\mathrm{d}x\) | dM1 |
| Obtains \(\tfrac{1}{2}v^2 = \mp\dfrac{32}{x}\ (+C)\) or equivalent e.g. \(\tfrac{0.1}{2}v^2 = -\dfrac{16}{5x}\ (+C)\) | A1 |
| Substituting \(x = 2\) if + used earlier or – 2 if – used in d.e. \(x = 2,\ v = \pm 8 \Rightarrow 32 = -16 + C \Rightarrow C = 48\) (or value appropriate to their correct equation) | M1 A1 |
| \(v = 0 \Rightarrow \dfrac{32}{x} = 48 \Rightarrow x = \tfrac{2}{3}\) m (N.B. \(-\tfrac{2}{3}\) is not acceptable for final answer) | M1 A1 cao |
| (8 marks) |
Notes
N.B \(\dfrac{\mathrm{d}}{\mathrm{d}x}\left(\tfrac{1}{2}mv^2\right) = \dfrac{16}{5x^2}\), is also a valid approach.
Last two method marks are independent of earlier marks and of each other