A2 October 2021 Q2
2. At time \(t = 0\), a small stone \(P\) of mass \(m\) is released from rest and falls vertically through the air. At time \(t\), the speed of \(P\) is \(v\) and the resistance to the motion of \(P\) from the air is modelled as a force of magnitude \(kv^2\), where \(k\) is a constant.
At time \(t\), \(P\) has fallen a distance \(s\).
| Scheme | Marks | AO |
|---|---|---|
| \(mg - kv^2 = m\dfrac{\mathrm{d}v}{\mathrm{d}t}\) | M1 | 2.5 |
| Separate variables and integrate | M1 | 2.1 |
| A correct equation in any form (ignore constant or limits) e.g. \(t = \dfrac{m}{k}\,\dfrac{1}{2\sqrt{\frac{mg}{k}}}\ln\left(\dfrac{\sqrt{\frac{mg}{k}} + v}{\sqrt{\frac{mg}{k}} - v}\right) \quad (+\,C)\) | A1 | 1.1b |
| \(t = \dfrac{V}{2g}\ln\left(\dfrac{V + v}{V - v}\right)\) where \(V^2 = \dfrac{mg}{k}\) * | A1* | 2.2a |
| (4) |
Notes
M1: Equation of motion with correct form for the acceleration
M1: Separate the variables and integrate (‘standard integral’)
A1: Correct equation in any form (ignoring constant or limits)
A1*: Correctly obtain the printed answer including dealing with constant or limits
| Scheme | Marks | AO |
|---|---|---|
| \(V^2 = \dfrac{mg}{k} \Rightarrow kV^2 = mg\) i.e. resistance \(=\) weight OR using answer to (a): As \(t \to \infty,\ v \to V\) from below | B1 | 1.1b |
| Hence \(V\) is the terminal velocity of the stone oe | B1 | 2.4 |
| (2) |
Notes
B1: Correctly rearrange and interpret OR correctly argue and interpret
B1: Correct statement or equivalent
| Scheme | Marks | AO |
|---|---|---|
| \(mg - kv^2 = mv\dfrac{\mathrm{d}v}{\mathrm{d}s}\) | M1 | 2.5 |
| Separate variables and integrate | M1 | 2.1 |
| \(s = -\dfrac{m}{2k}\ln\left(\dfrac{mg}{k} - v^2\right) \quad (+\,D)\) | A1 | 1.1b |
| \(s = \dfrac{V^2}{2g}\ln\left(\dfrac{V^2}{V^2 - v^2}\right)\) * | A1* | 2.2a |
| (4) | ||
| (10 marks) |
Notes
M1: Equation of motion with correct form for the acceleration
M1: Separate the variables and integrate (‘standard integral’)
A1: Correct equation in any form (ignoring constant or limits)
A1*: Correctly obtain the printed answer including dealing with constant or limits