Power: (b) P = Fv

The idea

Power is the rate of doing work or transferring energy. When a constant force F acts on an object moving at constant velocity v in the direction of the force, the mechanical power delivered is P = Fv. This follows from P = W/t and W = Fs, so P = Fs/t = Fv. For example, a car engine provides a forward force of 2000 N while travelling at 15 m s⁻¹, so the useful output power is P = 2000 × 15 = 30 000 W = 30 kW. If the force and velocity are not in the same direction, only the component of force along the velocity contributes, so P = Fv cos θ. In this course, you apply P = Fv to constant-speed situations and to instantaneous power when F and v are known at a particular moment.

What earns the marks

• Power is the rate of energy transfer or work done, measured in watts (W), where 1 W = 1 J s⁻¹.
• For a constant force F acting in the direction of motion at constant velocity v, the mechanical power is given by P = Fv.
• The equation is derived from P = W/t and W = Fs, giving P = Fs/t = Fv when v = s/t is constant.
• If force and velocity are not parallel, use the component of force along the velocity: P = Fv cos θ.
• Typical application: a vehicle moving at constant speed against resistive forces, where the driving force equals the total resistive force.

Common mistakes and how to put them right

• Mistaking power for force or energy.

  Put it right by using P = Fv only when a rate is required.

• Using P = Fv with v as an average speed when the force is not constant.

  Put it right by checking that F and v are constant or by using instantaneous values.

• Forgetting to convert units such as km h⁻¹ to m s⁻¹ before substituting into P = Fv.

  Put it right by converting all quantities to SI units first.

• Assuming P = Fv always gives the total power input.

  Put it right by distinguishing input power from useful output power.

Exam tips

• Always write the equation P = Fv, substitute values with units, and state the unit of power as W or J s⁻¹.
• If a velocity is given in km h⁻¹, convert to m s⁻¹ by dividing by 3.6 before using P = Fv.
• In multiple-choice questions, check whether the force and velocity are in the same direction; if not, use the component of force along the velocity.
• Link P = Fv to efficiency questions: useful output power may be less than input power, so calculate efficiency separately if needed.