A2 June 2022 Q4
4. The velocity \(v\,\mathrm{m\,s^{-1}}\), of a raindrop, \(t\) seconds after it falls from a cloud, is modelled by the differential equation
\[\frac{\mathrm{d}v}{\mathrm{d}t} = -0.1v^2 + 10 \qquad t \geqslant 0\]Initially the raindrop is at rest.
Given that the initial acceleration of the raindrop is found to be smaller than is suggested by the current model,
| Scheme | Marks | AO |
|---|---|---|
| Identifies \(t_0 = 0,\ v_0 = 0,\ \left(\dfrac{\mathrm{d}v}{\mathrm{d}t}\right)_{0} = 10\) and \(h = 0.5\) | B1 | 3.4 |
| \(v_1 = v_0 + h\left(\dfrac{\mathrm{d}v}{\mathrm{d}t}\right)_{0} \Rightarrow v_1 = 0 + 0.5 \times 10 = \ldots\) | M1 | 1.1b |
| \(v_1 = 5\) | A1 | 1.1b |
| \(\left(\dfrac{\mathrm{d}v}{\mathrm{d}t}\right)_{1} = -0.1(5)^2 + 10 = \ldots\{7.5\}\) \(v_2 = v_1 + h\left(\dfrac{\mathrm{d}v}{\mathrm{d}t}\right)_{1} \Rightarrow v_2 = 5 + 0.5 \times 7.5 = \ldots\) | M1 | 3.4 |
| \(v_2 = 8.75\) so \(8.75\,\mathrm{m\,s^{-1}}\) | A1 | 1.1b |
| (5) |
Notes
B1: Uses the model to identify the correct initial conditions and requirements for \(h\). May be implied by use in the equation.
M1: Applies the approximation formula with their values for \(v_0\), \(\left(\dfrac{\mathrm{d}v}{\mathrm{d}t}\right)_{0}\) and \(h\) to find a value for \(v_1\)
A1: \(v_1 = 5\)
M1: Uses their \(v_1\) to find a value for \(\left(\dfrac{\mathrm{d}v}{\mathrm{d}t}\right)_{1}\) and applies the approximation formula with their values for \(v_1\), \(\left(\dfrac{\mathrm{d}v}{\mathrm{d}t}\right)_{1}\) and \(h\) to find a value for \(v_2\)
A1: \(v_2 = 8.75\) or \(8.75\,\mathrm{m\,s^{-1}}\)
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{\mathrm{d}v}{\mathrm{d}t} = -0.1v^2 + A\) where \(0 \lt A \lt 10\) | B1 | 3.5c |
| (1) | ||
| (6 marks) |
Notes
B1: Reduce the value of 10 or explains this is what needs reducing, but do not accept 0 or negative values in place of the 10. Note: “change the 10” is B0 if it does not explain how to change it.