AS June 2018 Q4
4. A particle, \(P\), moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\) is \(v\ \text{m s}^{-1}\) in the direction of \(x\) increasing and the acceleration of \(P\) is \(a\ \text{m s}^{-2}\) in the direction of \(x\) increasing.
When \(t = 0\) the particle is at rest at the origin \(O\).
Given that \(a = \dfrac{5}{2}(5 - v)\)
At the instant when \(v = 2.5\), the particle is \(d\) metres from \(O\).
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle\int \dfrac{1}{4}\,\mathrm{d}t = \int \dfrac{1}{50 - 10v}\,\mathrm{d}v\) | M1 | 3.1a |
| \(\dfrac{1}{4}t = -\dfrac{1}{10}\ln(50 - 10v)\ (+C)\) | A1 | 1.1b |
| \(\dfrac{1}{4}t = -\dfrac{1}{10}\ln(50 - 10v) + \dfrac{1}{10}\ln 50\) | M1 | 1.1b |
| \(-\dfrac{5t}{2} = \ln\left(\dfrac{5 - v}{5}\right)\) | M1 | 1.1b |
| \(v = 5\left(1 - \mathrm{e}^{-2.5t}\right)\) * | A1* | 2.1 |
| (5) |
Notes
M1: Strategy to find \(v\) and attempt the integration
A1: Correct integration
M1: Use boundary conditions as limits or evaluate constant of integration in an expression involving \(\lambda\ln(a + bv)\) and \(\mu t\)
M1: Remove logarithm to express \(v\) in terms of \(t\)
A1*: Obtain given answer from correct working
| Scheme | Marks | AO |
|---|---|---|
| limiting value is 5 | B1 | 2.2a |
| (1) |
Notes
B1: Correct answer from correct working
| Scheme | Marks | AO |
|---|---|---|
| Equation in \(x\) and \(t\): \(\dfrac{\mathrm{d}x}{\mathrm{d}t} = 5\left(1 - \mathrm{e}^{-2.5t}\right)\) | M1 | 1.1a |
| \(\Rightarrow \displaystyle\int 1\,\mathrm{d}x = \int 5\left(1 - \mathrm{e}^{-2.5t}\right)\mathrm{d}t\) | M1 | 1.1b |
| \(x = 5t + 2\mathrm{e}^{-2.5t}\ (+C)\) | A1 | 1.1b |
| Use \(v = 2.5\) and \(v = 5\left(1 - \mathrm{e}^{-2.5t}\right)\) to find value of \(t\) | M1 | 3.1a |
| \(1 - \dfrac{2.5}{5} = \mathrm{e}^{-2.5t} \Rightarrow t = \dfrac{2}{5}\ln 2\) | A1 | 1.1b |
| \(\Big[x\Big]_0^d = \Big[5t + 2\mathrm{e}^{-2.5t}\Big]_0^{\frac{2}{5}\ln 2}\) | M1 | 2.1 |
| \(d = 2\ln 2 - 1\) * | A1* | 1.1b |
| (7) | ||
| (13 marks) |
Notes
M1: Set up equation of motion in terms of \(x\) and \(t\)
M1: Separate variables and attempt integration of both sides
A1: Any equivalent form. Condone if \(+C\) not seen
M1: Use \(v = 2.5\) to find limit for \(t\)
A1: Any equivalent exact form. (0.277)
M1: Use boundary conditions as limits or evaluate constant of integration in an expression involving \(\lambda t\) and \(\mu\mathrm{e}^{-2.5t}\)
A1*: Sufficient correct working to justify given answer