M3 June 2012 Q1
1. A particle \(P\) is moving along the positive \(x\)-axis. At time \(t = 0\), \(P\) is at the origin \(O\). At time \(t\) seconds, \(P\) is \(x\) metres from \(O\) and has velocity \(v = 2\mathrm{e}^{-x}\) m s\(^{-1}\) in the direction of \(x\) increasing.
(a) Find the acceleration of \(P\) in terms of \(x\). (3)
(b) Find \(x\) in terms of \(t\). (6)
| Scheme | Marks |
|---|---|
| Use of \(a = v\dfrac{\mathrm{d}v}{\mathrm{d}x}\) or \(a = \dfrac{\mathrm{d}}{\mathrm{d}x}\left(\dfrac{1}{2}v^2\right)\) | M1 |
| \(a = 2e^{-x}.-2e^{-x}\) or \(v^2 = 4e^{-2x}\) | A1 |
| \(a = -4e^{-2x}\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Separate the variables and attempt to integrate: | M1 |
| \(\displaystyle\int 2\,\mathrm{d}t = \int e^{x}\,\mathrm{d}x\) | |
| \(2t = e^{x} + C\) | A1A1 |
| \(t = 0,\ x = 0 \Rightarrow C = -1,\ \ 2t = e^{x} - 1\) | M1A1 |
| \(x = \ln\left(2t + 1\right)\) | A1 |
| (6) | |
| (9 marks) |