A2 June 2025 Q2
2. A van of mass 900 kg is moving along a straight horizontal road.
When the speed of the van is \(v\ \text{m s}^{-1}\), the resistance to the motion of the van is modelled as a force of magnitude \(50v\) N.
The engine of the van is working at a constant rate of 12 kW.
At the instant when the speed of the van is \(V\ \text{m s}^{-1}\), the acceleration of the van is \(0.2\ \text{m s}^{-2}\)
Using the model,
Later on, the van is moving up a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin\alpha = \dfrac{1}{12}\). When the speed of the van is \(v\ \text{m s}^{-1}\), the resistance to the motion of the van from non-gravitational forces is modelled as a force of magnitude \(50v\) N.
When the van has constant speed \(U\ \text{m s}^{-1}\), the engine of the van is working at a constant rate of 15 kW.
Using the model,
| Scheme | Marks | AO |
|---|---|---|
| Equation of motion of the van | M1 | 3.3 |
| \(F - 50V = 900 \times 0.2\) | A1 | 1.1b |
| Use of \(P = FV\) \(\left(\dfrac{12000}{V} - 50V = 900 \times 0.2\right)\) | M1 | 3.4 |
| \(\left(50V^2 + 180V - 12000 = 0\right)\) \((V =)\ 14\) or better (correctly rounded) | A1 | 1.1b |
| (4) |
Notes
M1: Form an equation of motion for the van to produce an equation in \(V\) only or form an equation of motion for the van to produce an equation in \(V\) and \(F\) only where \(F\) is the driving force of the van’s engine.
Must be dimensionally correct with required terms. Condone sign errors.
A1: Correct unsimplified equation.
M1: Use of \(P = FV\) where \(F\) is the driving force of the van’s engine. Implied by use of \(\dfrac{12000}{V}\) with their equation of motion. Condone use of 12 or a slip with the number of zeros for this M mark.
A1: Obtain only 14 or better (correctly rounded). Calculator display gives 13.79615337.
If a negative value is seen, it must be rejected.
| Scheme | Marks | AO |
|---|---|---|
| Equation of motion of the van | M1 | 3.3 |
| \(F - 50U - 900g\sin\alpha = 0\) \(\dfrac{15000}{U} - 50U - 75g = 0\) | A1 A1 | 1.1b 1.1b |
| \(\left(50U^2 + 735U - 15000 = 0\right)\) \((U =)\ 11\) or 11.5 | A1 | 1.1b |
| (4) | ||
| (8 marks) |
Notes
M1: Form equation of motion for the van with \(a = 0\) to produce an equation in \(U\) only or in \(U\) and \(F\) only. Must be dimensionally correct with required terms. Condone sign errors and sin/cos confusion. Condone use of another letter instead of \(U\) throughout. Condone use of 15 or a slip with the number of zeros.
M0 if the answer from (a) is used in (b). M0 for use of \(P = 12\,000\)
A1: Unsimplified equation in \(F\) and \(U\) (or \(U\) only) with at most one error. Trig and \(F\) do not need not be replaced.
A1: Correct unsimplified equation in \(U\) only (trig and \(F\) replaced).
A1: Obtain only 11 or 11.5 (2 or 3sf after use of 9.8). If a negative value is seen, it must be rejected. A0 if the answer follows use of \(g = 9.81\ \text{m s}^{-2}\)