A2 October 2021 Q1
1. A van of mass 900 kg is moving along a straight horizontal road.
At the instant 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 \((500 + 7v)\) N.
When the engine of the van is working at a constant rate of 18 kW, the van is moving along the road at a constant speed \(V\ \text{m s}^{-1}\)
Later on, the van is moving up a straight road that is inclined to the horizontal at an angle \(\theta\), where \(\sin\theta = \dfrac{1}{21}\)
At the instant 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 \((500 + 7v)\) N.
The engine of the van is again working at a constant rate of 18 kW.
| Scheme | Marks | AO |
|---|---|---|
| Equation of motion: \(F = 500 + 7V\) | M1 | 3.3 |
| Use of \(18000 = F \times V\) | M1 | 3.4 |
| \(\Rightarrow \dfrac{18000}{V} = 500 + 7V\) | A1 | 1.1b |
| \(\Rightarrow 7V^2 + 500V - 18000 = 0\) | M1 | 1.1b |
| \(V = 26\ \ (26.309\ldots)\) | A1 | 1.1b |
| (5) |
Notes
M1: Dimensionally correct. Condone sign errors. Must be using \(a = 0\)
M1: Correct use of \(P = Fv\)
A1: Correct unsimplified equation. Allow with \(F\). Allow with 18K
M1: Form and solve a 3 term quadratic
A1: 26 or better (26.309……)
| Scheme | Marks | AO |
|---|---|---|
| Equation of motion: | M1 | 3.3 |
| \(\dfrac{18000}{15} - (500 + 7 \times 15) - 900g \times \dfrac{1}{21} = 900a\) | A1 A1 | 1.1b 1.1b |
| \(a = 0.194\ (0.19)\ (\text{m s}^{-2})\) | A1 | 1.1b |
| (4) | ||
| (9 marks) |
Notes
M1: Dimensionally correct. All terms required. Condone sign errors and sin/cos confusion. Omission of \(g\) is an accuracy error
A1 A1: Unsimplified equation with at most one error
Correct unsimplified equation. Allow if \(\sin\theta\) not substituted. Allow with 18K
A1: 2 sf or 3 sf only not \(\tfrac{7}{36}\)