Power and Efficiency

1. Definition

Power

Power is the rate of doing work, or the rate of energy transfer.

P = W/t = E/t

• P is power in watts (W), W is work done in joules (J), E is energy transferred in joules (J), and t is time in seconds (s).
• 1 W = 1 J s⁻¹
• Power is a scalar quantity.

Review Energy calculations first if finding Eₖ, Δ Eₚ or dissipated energy still feels difficult.

Power is the gradient of an energy–time graph

Two straight lines on an energy-time graph: the steeper line represents higher power (faster energy transfer).

Scroll across the graph to read all labels.

Open full-size graph

View figure data

Values for Power is the gradient of an energy–time graph

Time (s)	100 W device	200 W device
0	0	0
5	500	1000

Values for Power is the gradient of an energy–time graph

Time (s)	100 W device	200 W device
0	0	0
5	500	1000

For the steeper line, gradient = (1000 - 0)/(5.0 - 0) = 200 J s⁻¹ = 200 W

The gradient unit confirms that the graph gives a rate of energy transfer.

2. Key Ideas

• Always convert time to seconds before using P = E/t.
• A device has higher power if it transfers the same energy in less time, or more energy in the same time.
• If a load is lifted at constant speed, the useful work done is the gain in GPE: W_useful = mgh (or Fh if weight F is given).
• Efficiency, η, compares the useful output with the total input. Use either energy values, or power values measured over the same interval, but do not mix them: η = E_useful/Eᵢₙₚᵤₜ × 100% η = P_useful/Pᵢₙₚᵤₜ × 100%
• Efficiency cannot exceed 100%.
• High power does not necessarily mean high efficiency: power describes how fast energy is transferred, while efficiency describes the useful fraction.

3. Detailed Explanations

Calculating power

To find power:

1. Find the energy transferred E (or work done W).
2. Convert time to seconds.
3. Use P = E/t (or P = W/t).

For example, if 600 J is transferred in 3.0 s, P = 600/3.0 = 200 W

Efficiency

Energy is conserved, but not all input energy is transferred usefully. For a device:

total input energy = useful energy output + energy dissipated to the surroundings

The dissipated share commonly increases internal energy stores in the device and surroundings. Sound waves may also carry energy away, often because of friction, vibration or electrical resistance.

Efficiency of an energy transfer is:

η = E_useful/Eᵢₙₚᵤₜ × 100%

4. Common Mistakes

• Not converting minutes/hours to seconds.
• Mixing up power (W) and energy (J).
• Writing P = W/t but using W as “watt” instead of work done (J).
• Getting efficiency > 100% (usually input/output swapped).
• Forgetting to multiply by 100% when the answer is required as a percentage.
• Assuming the device with the greatest power is automatically the most efficient.
• Using useful energy over input power, or useful power over input energy. Both parts of an efficiency ratio must be the same type of quantity.

5. Exam Tips

• Start with the definition: power = energy transfer per unit time.
• Show unit conversions clearly (e.g. 10 min = 600 s).
• When comparing two power outputs, check what is the same:
  • same work but different time → compare time
  • same time but different work → compare work
• For efficiency questions, write input = useful + dissipated before substituting numbers.
• Convert a percentage efficiency to a decimal before rearranging an equation: 20% = 0.20.
• Sense-check the result: useful output cannot exceed total input, and efficiency cannot exceed 100%.

6. Worked Examples

7. Mind Stretchers