A2 June 2023 Q2
2. A car of mass 1000 kg moves in a straight line along a horizontal road at a constant speed \(U\ \text{m s}^{-1}\). The resistance to the motion of the car is a constant force of magnitude 400 N.
The engine of the car is working at a constant rate of 16 kW.
The car now pulls a trailer of mass 600 kg in a straight line along the road using a tow rope which is parallel to the direction of motion. The resistance to the motion of the car is again a constant force of magnitude 400 N. The resistance to the motion of the trailer is a constant force of magnitude 300 N.
The engine of the car is working at a constant rate of 16 kW.
The tow rope is modelled as being light and inextensible.
Using the model,
| Scheme | Marks | AO |
|---|---|---|
| \(F = \dfrac{16000}{v}\) | M1 | 3.3 |
| Equation of motion: \(F - 400 = 0\) | M1 | 3.1b |
| \(U = 40\) | A1 | 1.1b |
| (3) |
Notes
M1: Correct use of \(P = Fv\). The expression \(\dfrac{16000}{v}\) may be on a diagram or embedded in their \(F = ma\). Condone use of 16 000 or 16 for the method mark.
M1: Correct unsimplified equation of motion with \(a = 0\) or equilibrium equation. \(F\) does not need to be substituted.
A1: cao
| Scheme | Marks | AO |
|---|---|---|
| \(F = \dfrac{16000}{\left(\frac{20}{3}\right)}\) | M1 | 3.3 |
| Equation of motion for system or car or trailer: | M1 | 3.1b |
| \(F - 700 = 1600a\) or \(F - 400 - T = 1000a\) or \(T - 300 = 600a\) | A1 | 1.1b |
| Second equation of motion | A1 | 1.1b |
| \(T = 940\) or \(938\) or \(937.5\) or \(\dfrac{1875}{2}\) oe (N) | A1 | 1.1b |
| (5) | ||
| (8 marks) |
Notes
M1: Correct use of \(P = Fv\) with \(v = \dfrac{20}{3}\). This expression may be on the diagram or embedded in their \(F = ma\). Condone use of 16 000 or 16 for the method mark.
M1: An equation of motion for the whole system or car or trailer. Must have all terms and be dimensionally correct. Condone sign errors. M0 if \(a = 0\) is used.
NB: Full marks in (b) can be scored if consistent extra \(g\)’s (must be present in both ‘\(ma\)’ terms in a complete solution). Otherwise penalise as A error.
A1: One correct unsimplified equation.
A1: Two correct unsimplified equations.
Note: \(a = \dfrac{17}{16}\) but does not need to be seen.
A1: Correct answer