A LevelFurther Maths: Further Mechanics 1EdexcelRevision NotesWork, Energy & PowerWork, Energy & PowerPower

Power (Edexcel A Level Further Maths: Further Mechanics 1)

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Jamie W

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Jamie W

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Power

What is power?

  • Power is developed as an engine or machine does work
  • Power is the rate of doing work, usually by the driving force of an engine
    • It is the same as work done per second
    • The work done is converting fuel (a type of chemical energy) into a driving force
  • Power is a scalar quantity and can only take a positive value

How is power calculated?

  • Power is the rate of doing work and can be calculated by the formula

P space equals space fraction numerator W D over denominator t end fraction

Where WD is the work done by the driving force in Joules and t is the time in seconds

  • If a driving force, F N, is acting on an object moving with velocity, v m s-1 then power can be calculated using the formula

P space equals F v

    • The velocity must be in the same direction as the force
    • When labelling forces on a diagram, the rearrangement F equals P over v can be useful to write the force generated by a particular power, at a particular speed
    • For a constant velocity and a constant force, the above formula can be derived by recalling the formula

v space equals space s over t

                      where v is speed or velocity and s is distance or displacement.

deriving P=fv from definition of power being the rate at which work is done

    • The derivation can also be shown for a constant force, F N and considering the rate of doing work over a very small interval of time
      • P space equals space F fraction numerator delta s over denominator delta t end fraction
      • As the time δt gets smaller and approaches  fraction numerator d s over denominator d t end fractionthen
      • P space equals F fraction numerator d s over denominator d t end fraction equals F v 
  • The units for power are watts (W) or kilowatts (kW)
    • 1 watt = 1 joule per second
    • 1 kilowatt = 1000 watts = 1000 joules per second

How is power used in calculating maximum speed?

  • If the power is constant, then as a vehicle gains speed (v increases), the driving force, F, must decrease
    • This can be observed by considering P equals F v 
  • If the vehicle maintains maximum power, its driving force will decrease as its speed increases and it will eventually reach a point where its resultant force is zero (the driving force will balance the resistance forces)
    • When the resultant force becomes zero, by Newton’s Second Law, acceleration will also become zero and the vehicle will be travelling at maximum speed
    • The maximum power output of a vehicle can be found when the vehicle is moving at its maximum speed and the acceleration is zero
    • If the maximum power output is known, then maximum speed can be found when the vehicle is travelling at its maximum power with zero acceleration and the resultant forces are balanced

Exam Tip

  • Make sure you are using the correct force in your calculation, power is only generated by the driving force of an engine and so only this force should be used in the formula.
  • Always draw a diagram and add the forces. If the question involves an inclined slope remember to resolve the weight into components parallel and perpendicular to the slope first.
  • Remember to check the units carefully, power questions could be given in watts or kilowatts. It is also important to give your answer in the correct units, or if not specified, choose the most appropriate units for the question.
  • Exam questions can say "an engine works at a rate of..." to mean "an engine has a power of"

Worked example

A car of mass 1300 kg, including the driver, moves forwards on a straight horizontal road. There is a constant resistive force of 900 N acting on the car. Its maximum possible speed is 40 m s-1  Calculate the maximum power that the engine of the car can produce.

3-2-1-m2-power-worked-example

Worked example

A car of mass 1 tonne is moving at a constant speed of 9 ms-1 up a straight road inclined at 9° to the horizontal. The engine is working at a rate of 22 kW. Find the magnitude of the non-gravitational resistance to motion, to 3 significant figures.

worked example showing how to solve a problem using power (first part)worked example showing how to solve a problem using power (second part)

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    Jamie W

    Author: Jamie W

    Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.