AS October 2020 Q2
2. A car of mass 1000 kg moves along a straight horizontal road.
In all circumstances, when the speed of the car is \(v\ \text{m s}^{-1}\), the resistance to the motion of the car is modelled as a force of magnitude \(cv^2\) N, where \(c\) is a constant.
The maximum power that can be developed by the engine of the car is 50 kW.
At the instant when the speed of the car is \(72\ \text{km h}^{-1}\) and the engine is working at its maximum power, the acceleration of the car is \(2.25\ \text{m s}^{-2}\)
The maximum speed of the car when the engine is working at its maximum power is \(V\ \text{km h}^{-1}\).
| Scheme | Marks | AO |
|---|---|---|
| \(72\ \text{km h}^{-1} = 20\ \text{m s}^{-1}\) | B1 | 1.1b |
| (1) |
Notes
B1: \(20\ \text{m s}^{-1}\) seen
| Scheme | Marks | AO |
|---|---|---|
| Use of \(F = \dfrac{P}{v}\) and using the model | M1 | 3.4 |
| Equation of motion and using the model to form equation in \(c\) | M1 | 3.1b |
| \(\dfrac{50000}{20} - c \times 20^2 = 1000 \times 2.25 \qquad \left(c = \dfrac{5}{8}\right)\) | A1ft | 1.1b |
| Equation of motion and using the model | M1 | 3.1b |
| \(\dfrac{50000}{40} - c \times 40^2 = 1000a\) | A1ft | 1.1b |
| Solve for \(a\) | M1 | 1.1b |
| \(0.25\ (\text{m s}^{-2})\) | A1 | 1.1b |
| (7) |
Notes
M1: Follow through the 72 or their \(v\). Allow for 144 or their 144
M1: Correct no. of terms required
A1ft: Correct unsimplified equation ft on their 20
M1: Correct no. of terms required
Allow the second and third M marks if they have an equation in \(F\) rather than \(P\).
A1ft: Correct equation ft on their 40 and their \(c\)
M1: Complete method to solve for \(a\)
A1: Cao \(\left(\text{Accept } \dfrac{1}{4}\right)\)
| Scheme | Marks | AO |
|---|---|---|
| Equation of motion horizontally and using the model | M1 | 3.1b |
| \(\dfrac{50000}{W} - \dfrac{5}{8}W^2 = 0\) (max speed is \(W\ \text{m s}^{-1}\)) | A1ft | 1.1b |
| Solve for \(W\) and convert to \(\text{km h}^{-1}\) \((W = 43.088...)\) | M1 | 1.1b |
| \(V = 155\) (nearest whole number) | A1 | 1.1b |
| (4) | ||
| (12 marks) |
Notes
M1: Equation with correct no. of terms, correct structure and in terms of \(W\) only.
A1ft: Correct equation, ft on their \(c\) from part (b).
M1: Complete method to solve for \(V\) (including clear attempt to convert units)
A1: Cao (The Q asks for a whole number)