M3 June 2008 Q6
6. A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis. It moves away from the origin \(O\) under the action of a single force directed away from \(O\). When \(OP = x\) metres, the magnitude of the force is \(\dfrac{3}{(x + 1)^3}\) N and the speed of \(P\) is \(v\) m s\(^{-1}\). Initially \(P\) is at rest at \(O\).
(a) Show that \(v^2 = 6\left(1 - \dfrac{1}{(x + 1)^2}\right)\). (6)
(b) Show that the speed of \(P\) never reaches \(\sqrt{6}\) m s\(^{-1}\). (1)
(c) Find \(x\) when \(P\) has been moving for 2 seconds. (7)
| Scheme | Marks |
|---|---|
| \(F = ma\ (\rightarrow)\) \(\dfrac{3}{(x + 1)^3} = 0.5\text{a} = 0.5v\dfrac{\mathrm{d}v}{\mathrm{d}x}\) | M1A1 |
| \(\displaystyle\int \dfrac{3}{(x + 1)^3}\,\mathrm{d}x = 0.5\int v\,\mathrm{d}v\) Separate and \(\int\) | M1 |
| \(-\dfrac{3}{2(x + 1)^2} = \dfrac{1}{4}v^2\ (+\ c)\) | A1 |
| \(x = 0,\ v = 0 \Rightarrow c' = -\dfrac{3}{2}\) \(\therefore v^2 = 6\left(1 - \dfrac{1}{(x + 1)^2}\right)\) * | M1A1 cso |
| (6) |
| Scheme | Marks |
|---|---|
| \(\forall x \quad v^2 < 6 \quad \therefore v < \sqrt{6}\) \((\because (x + 1)^2\) always \(> 0)\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(v = \dfrac{\mathrm{d}x}{\mathrm{d}t} = \dfrac{\sqrt{6}\sqrt{(x + 1)^2 - 1}}{x + 1}\) | M1 |
| \(\displaystyle\int \dfrac{x + 1}{\sqrt{(x + 1)^2 - 1}}\,\mathrm{d}x = \sqrt{6}\int \mathrm{d}t\) | M1 |
| \(\sqrt{(x + 1)^2 - 1} = \sqrt{6}t + c'\) | M1 A1 |
| \(t = 0,\ x = 0 \Rightarrow c' = 0\) | M1 |
| \(t = 2 \Rightarrow (x + 1)^2 - 1 = (2\sqrt{6})^2\) | M1 |
| \((x + 1)^2 = 25 \quad \Rightarrow x = 4\) (\(c'\) need not have been found) | A1 cao |
| (7) | |
| (14 marks) |