M2 June 2016 Q2
2. A car of mass 800 kg is moving on a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = \dfrac{1}{20}\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving up the road at a constant speed of 12.5 m s\(^{-1}\), the engine of the car is working at a constant rate of \(3P\) watts. When the car is moving down the road at a constant speed of 12.5 m s\(^{-1}\), the engine of the car is working at a constant rate of \(P\) watts.
When the car is moving up the road at 12.5 m s\(^{-1}\) the engine is switched off and the car comes to rest, without braking, in a distance \(d\) metres. The resistance to the motion of the car from non-gravitational forces is still modelled as a constant force of magnitude \(R\) newtons.
| Scheme | Marks |
|---|---|
| M1 | |
| \(F = mg\sin\theta + R\ \ \left(F = R + 392\right)\) | A1 |
| \(G + mg\sin\theta = R\ \ \left(G = R - 392\right)\) | A1 |
| \(F = \dfrac{3P}{12.5}\) or \(G = \dfrac{P}{12.5}\) \(\Rightarrow \dfrac{3P}{12.5} = 392 + R\) or \(\dfrac{P}{12.5} = R - 392\) | B1 |
| \(\dfrac{2P}{12.5} = 2 \times 392,\ \ 2R = \dfrac{4P}{12.5}\) | M1 |
| \(P = 4900\ (500g),\ \ \ \ R = 784\ (80g)\) | A1 |
| (6) |
Notes
M1 Equation of motion up or down the road. Requires all 3 terms. Condone sign errors and trig confusion. Must be dimensionally correct.
A1 Correct equation up the road
A1 Correct equation down the road
B1 Use of \(F = \dfrac{P}{v}\) at least once
M1 Solve simultaneous equations for \(P\) or \(R\), provided \(F \neq G\) and \(P\) and \(3P\) used correctly
A1 CSO. Both values correct. Accept 2sf, 3sf or an exact multiple of g
| Scheme | Marks |
|---|---|
| Must be using work-energy. | |
| KE lost = PE gained + WD against R | M1 |
| \(\dfrac{1}{2} \times 800 \times 12.5^2\) \(= 800 \times 9.8 \times \dfrac{d}{20} + (\text{their } R) \times d\) | A1 A1ft |
| \(d = \dfrac{62500}{1176} = 53.1\) (m) | A1 |
| (4) | |
| (10 marks) |
Notes
M1 Equation needs all 3 terms and no extras. Condone sign errors.
A1 At most 1 error. Allow with \(R\) (with trig. substituted) \(\left(62500 = 392d + Rd\right)\)
A1ft Correct equation in their \(R\) (with trig. substituted)
A1 CSO. Accept 53(m)