AS June 2018 Q3
3. A van of mass 750 kg is moving along a straight horizontal road. At the instant when the van is moving at \(v\ \text{m s}^{-1}\), the resistance to the motion of the van is modelled as a force of magnitude \(\lambda v\) N, where \(\lambda\) is a constant.
The engine of the van is working at a constant rate of 18 kW.
At the instant when \(v = 15\), the acceleration of the van is \(0.6\ \text{m s}^{-2}\)
The van now moves up a straight road inclined at an angle to the horizontal, where \(\sin\alpha = \dfrac{1}{15}\)
At the instant when the van is moving at \(v\ \text{m s}^{-1}\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude \(50v\) N.
When the engine of the van is working at a constant rate of 12 kW, the van is moving at a constant speed \(V\ \text{m s}^{-1}\)
| Scheme | Marks | AO |
|---|---|---|
| Use of \(P = Fv\) | B1 | 1.1a |
| Equation of motion: \(F - \lambda v = 750 \times 0.6\) | M1 | 2.1 |
| \(\dfrac{18000}{15} - \lambda \times 15 = 750 \times 0.6\) | A1 | 1.1b |
| \(1200 - 15\lambda = 450 \Rightarrow \lambda = 50\) * | A1* | 1.1b |
| (4) |
Notes
B1: Use of \(P = Fv\) seen or implied. Allow in (b) if not seen in (a)
M1: Requires all three terms. Must be dimensionally correct.
Need not have substituted for \(F\). Condone sign errors.
Allow if equation not seen but all steps in working correct.
The method needs to show that \(\lambda = 50\) is the only solution.
A1: Correct unsimplified equation
A1*: Obtain given answer correctly
| Scheme | Marks | AO |
|---|---|---|
| Overall strategy | M1 | 3.1b |
| Equation of motion | M1 | 3.4 |
| \(\dfrac{12000}{V} - 50V - 750g\sin\alpha = 0\) | A1 | 1.1b |
| \(\dfrac{12000}{V} - 50V - 490 = 0 \Rightarrow 5V^2 + 49V - 1200 = 0\) | A1 | 1.1b |
| \(\Rightarrow V \left(= \dfrac{-49 + \sqrt{49^2 + 20 \times 1200}}{10}\right) = 11.3\) only | A1 | 1.1b |
| (5) | ||
| (9 marks) |
Notes
M1: Complete strategy e.g. use the model to form quadratic in \(V\) and solve for \(V\)
M1: Use the model to form equation of motion. All terms required.
Condone sign errors and sin/cos confusion.
Need not have substituted for \(F\).
A1: Substituted equation with at most one error (unsimplified). Allow in \(F\) or \(V\).
A1: Correct quadratic equation. e.g. \(5V^2 + 49V - 1200 = 0\) or equivalent
Allow in \(F\) or \(V\).
A1: Accept 11 or 11.3 (follows use of 9.8)
Negative root should be rejected if seen