All subjects · Mathematics · WME02 Mechanics 2

Power: driving force, maximum speed and acceleration

Edexcel IAL Mathematics · WME02 Mechanics 2

Power is the rate of doing work. For a vehicle, power links the engine's driving force to the speed, and this gives a family of standard M2 questions about maximum speed and acceleration.

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Power

Power = work done ÷ time, measured in watts (W); 1 kW = 1000 W.

For a force F moving at speed v in its direction, P = Fv. So the driving force is F = P/v. At a fixed power, the faster the vehicle goes, the smaller the driving force.

Maximum speed

A vehicle at its maximum speed is not accelerating, so the driving force equals the total resistance: P/v = R, giving v_max = P/R.

Acceleration at a given speed

Below maximum speed, use Newton's second law along the road: P/v - R = ma. If the resistance depends on the speed (for example R = kv), find k from one situation and use it in another.

Work done by an engine working at constant power P for time t is Pt.

Worked example

A car of mass 1000 kg has an engine working at 30 kW. The resistance to motion is 1200 N on a level road. Find its maximum speed and its acceleration at 15 m/s.

  1. Maximum speed: 30 000/v = 1200, so v = 25 m/s.
  2. At 15 m/s: F = 30 000/15 = 2000 N.
  3. 2000 - 1200 = 1000a, so a = 0.8 m/s2.

Answer: 25 m/s; 0.8 m/s2

Key idea: P = Fv, so F = P/v. Maximum speed: P/v = R. Otherwise P/v - R = ma.

Check yourself

  1. Find the power, in kW, needed to drive a vehicle at a constant 20 m/s against a resistance of 800 N.
    Show answer

    16 kW. At constant speed F = 800 N, so P = 800 × 20 = 16 000 W = 16 kW.

  2. A car's engine works at 24 kW against a resistance of 600 N on a level road. Find its maximum speed.
    Show answer

    40 m/s. 24 000/v = 600, v = 40 m/s.

  3. A car of mass 1200 kg works at 18 kW against a resistance of 500 N on a level road. Find its acceleration when its speed is 12 m/s, to 3 significant figures.
    Show answer

    0.833 m/s2. F = 18 000/12 = 1500 N. 1500 - 500 = 1200a, a = 0.833 m/s2.

  4. A car of mass 1000 kg has resistance kv N at speed v m/s. Working at 36 kW, its maximum speed is 30 m/s. Find its acceleration at 20 m/s with the same power.
    Show answer

    1 m/s2. At 30 m/s: 36 000/30 = 1200 = 30k, so k = 40. At 20 m/s: F = 1800, R = 800, so a = 1000/1000 = 1 m/s2.

  5. An engine works at a constant 15 kW for 1 minute. Find the work it does, in kJ.
    Show answer

    900 kJ. W = Pt = 15 000 × 60 = 900 000 J = 900 kJ.

  6. A van of mass 800 kg moves on a level road against a resistance of 400 N. Find the power, in kW, needed to give it an acceleration of 0.5 m/s2 when its speed is 10 m/s.
    Show answer

    8 kW. F - 400 = 800 × 0.5, so F = 800 N. P = 800 × 10 = 8000 W = 8 kW.

  7. A vehicle working at constant power reaches its maximum speed. What is its acceleration then?
    • g
    • P/m
    • Zero, because the driving force equals the resistance
    • It is greatest at maximum speed
    Show answer

    Zero, because the driving force equals the resistance. At maximum speed there is no resultant force.

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Last updated 8 November 2026. Spotted a mistake? Tell us.