A-Level Mathematics May/June 2024 Q4: A car of mass 1700kg is pulling a trailer of mass 300kg along a straight horizontal roa…

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  Let the driving force of the car be $DF$, the acceleration of the system be $a$, and the tension in the cable be $T$.

  The car and trailer move together, so they have the same speed and acceleration.

  First, calculate the driving force from the power and speed. The relationship is Power = Driving Force × speed ($P = DF \times v$). $14000 = DF \times 20$ $DF = \frac{14000}{20} = 700$ N

  Next, apply Newton's Second Law ($F_{net} = ma$) to the entire system (car and trailer). The total mass is $m_{total} = 1700 + 300 = 2000$ kg. The total resistance is $R_{total} = 400 + 150 = 550$ N.

  For the whole system: $DF - R_{total} = m_{total} \times a$ $700 - 550 = 2000a$ $150 = 2000a$ $a = \frac{150}{2000} = \frac{3}{40} = 0.075$ m/s²

  Now, to find the tension, apply Newton's Second Law to the trailer alone. The forces acting on the trailer are the tension $T$ pulling it forward and the resistance $R_T = 150$ N opposing the motion.

  For the trailer: $T - R_T = m_T \times a$ $T - 150 = 300 \times 0.075$ $T - 150 = 22.5$ $T = 150 + 22.5$ $T = 172.5$ N

  Alternatively, we could consider the car alone: $DF - R_C - T = m_C \times a$ $700 - 400 - T = 1700 \times 0.075$ $300 - T = 127.5$ $T = 300 - 127.5 = 172.5$ N

  Final Answer: The acceleration is $0.075$ m/s² and the tension is $172.5$ N.

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  How the marks are awarded

  • B1 — Correctly calculating the driving force as 700 N using the formula $P = DF \times v$.
  • M1 — Applying Newton's Second Law to the whole system or one of the individual parts, with the correct number of forces included (e.g., $DF - R_{total} = m_{total}a$ or $T - R_T = m_T a$).
  • A2 — Forming two correct equations of motion. The model answer shows the equation for the whole system ($700 - 550 = 2000a$) and for the trailer ($T - 150 = 300a$). One correct equation earns A1.
  • DM1 — Solving the valid equations to find a numerical value for both acceleration and tension. This mark is dependent on the M1 mark.
  • A1 — Stating the correct final value for the tension, $T = 172.5$ N. The correct acceleration ($a=0.075$ m/s²) must also be found.

  Common mistakes

  • Using the power (14000 W) directly as the driving force in the $F=ma$ equation, instead of first calculating the force using $DF = P/v$.
  • Making sign errors in the equations of motion, such as adding a resistance force instead of subtracting it.
  • When considering the car alone, forgetting to include the tension force from the cable pulling backwards on the car.
  • Applying Newton's Second Law to the whole system but forgetting to include the resistance on the trailer (or the car).

  Examiner tip: For connected particle problems, always consider applying Newton's Second Law to the system as a whole and to each particle individually to generate the necessary equations.

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