M4 June 2017 Q3
3. A cyclist and her bicycle have a combined mass of 75 kg. The cyclist travels along a straight horizontal road. The cyclist produces a constant driving force of magnitude 150 N. At time \(t\) seconds, the speed of the cyclist is \(v\) m s\(^{-1}\), where \(v \lt \sqrt{50}\). As the cyclist moves, the total resistance to motion of the cyclist and her bicycle has magnitude \(3v^2\) newtons. The cyclist starts from rest. At time \(t\) seconds, she has travelled a distance \(x\) metres from her starting point.
Find
| Scheme | Marks |
|---|---|
| M1 | |
| \(75v\dfrac{\mathrm{d}v}{\mathrm{d}x} = 150 - 3v^2\) | A1 |
| \(\displaystyle\int\dfrac{75v}{150 - 3v^2}\,\mathrm{d}v = \int 1\,\mathrm{d}x\) | M1 |
| \(x = \left[-\dfrac{25}{2}\ln\left(50 - v^2\right)\right]_0^v\) | DM1 |
| \(= -\dfrac{25}{2}\ln\left(\dfrac{50 - v^2}{50}\right)\) | A1 |
| \(-\dfrac{2x}{25} = \ln\left(1 - \dfrac{v^2}{50}\right),\ v^2 = 50\left(1 - \mathrm{e}^{\frac{-2x}{25}}\right)\) | DM1 |
| \(v = \sqrt{50\left(1 - \mathrm{e}^{\frac{-2x}{25}}\right)}\) | A1 |
| (7) |
Notes
M1 Differential equation in \(v\) and \(x\). No additional/missing terms. Condone sign error(s)
M1 Separate variables
DM1 Integrate and use limits
DM1 Change the subject to \(v\) or \(v^2\)
3a alt
| \(\dfrac{\mathrm{d}v^2}{\mathrm{d}x} + \dfrac{2}{25}v^2 = 4\) | M1A1 |
| Integrating factor: \(\mathrm{e}^{\frac{2}{25}x}\) | M1 |
| \(v^2\mathrm{e}^{\frac{2}{25}x} = \displaystyle\int 4\mathrm{e}^{\frac{2}{25}x}\,\mathrm{d}x = 50\mathrm{e}^{\frac{2}{25}x}\ (+C)\) \(v = 0, x = 0 \Rightarrow C = -50\) | M1A1 |
| \(v^2 = 50 - 50\mathrm{e}^{-\frac{2}{25}x}\) | DM1 |
| \(v = \sqrt{50\left(1 - \mathrm{e}^{\frac{-2x}{25}}\right)}\) | A1 |
| (7) |
M1A1 Integrate and use limits
DM1 Change the subject to \(v\) or \(v^2\)
| Scheme | Marks |
|---|---|
| \(75\dfrac{\mathrm{d}v}{\mathrm{d}t} = 150 - 3v^2\) | M1 |
| \(\displaystyle\int\dfrac{75}{150 - 3v^2}\,\mathrm{d}v = \int 1\,\mathrm{d}t\) | M1 |
| \(t = \displaystyle\int\dfrac{25}{50 - v^2}\,\mathrm{d}v\) \(= \dfrac{25}{2\sqrt{50}}\displaystyle\int\dfrac{1}{\sqrt{50} + v} + \dfrac{1}{\sqrt{50} - v}\,\mathrm{d}v\) | |
| \(= \dfrac{25}{2\sqrt{50}}\left(\ln\left(\sqrt{50} + v\right) - \ln\left(\sqrt{50} - v\right)\right)\) | A1 |
| \(t = \dfrac{25}{2\sqrt{50}}\ln\left(\dfrac{\sqrt{50} + v}{\sqrt{50} - v}\right) - \dfrac{25}{2\sqrt{50}}\ln\left(\dfrac{\sqrt{50}}{\sqrt{50}}\right)\) | DM1 |
| \(= \dfrac{25}{2\sqrt{50}}\ln\left(\dfrac{\sqrt{50} + v}{\sqrt{50} - v}\right)\) | A1 |
| (5) | |
| (12 marks) |
Notes
M1 Differential equation in \(v\) and \(t\). No additional/missing terms. Condone sign error(s)
M1 Separate variables and integrate
A1 With or without constant of integration. Or \(\dfrac{25}{\sqrt{50}}\operatorname{arc\,tanh}\dfrac{v}{\sqrt{50}}\)
DM1 Use limits 0 and \(v\)
A1 \(\left(= \dfrac{5\sqrt{2}}{4}\ln\left(\dfrac{\sqrt{50} + v}{\sqrt{50} - v}\right)\right)\) or equivalent