# Power

> CIE A-Level Physics · Work, energy and power
> Source: https://www.owlsprep.com/study/cie-9702-u5-power/

This sub-topic covers the definition of power, relationships between power, work and energy, calculations for mechanical and electrical systems, and the link between power and efficiency, which is commonly tested in both multiple choice and structured questions.

**Prerequisites:** [Work done and energy transfer](https://www.owlsprep.com/study/cie-9702-u5-work-energy-transfer/)

## Learning objectives

- Define power as the rate of work done or energy transfer
- Calculate power for mechanical and electrical systems
- Derive and use the relationship $P=Fv$ for constant force
- Solve problems involving efficiency and power

## Definition of Power

**Power** — Power is defined as the rate of doing work or the rate of energy transfer. The SI unit of power is the watt ($\text{W}$), where 1 watt = 1 joule per second.

*Notation:* $P$

*Example:* A 60 W light bulb transfers 60 J of electrical energy to heat and light every second.

Average power over a time interval $\Delta t$ is calculated as:

$$P_{average} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$$

**Worked example:** A weightlifter lifts a 100 kg barbell 2 m above the ground in 2.5 seconds. What is their average power output?

1. First calculate work done to lift the barbell: $W = mgh = 100 \times 9.81 \times 2 = 1962 \text{ J}$
2. Substitute into average power formula:
3. $$P = \frac{W}{\Delta t} = \frac{1962}{2.5} = 784.8 \approx 780 \text{ W}$$

> **Exam tip:** Always check that your answer has the correct units: power is always in watts, not joules.

## Mechanical Power for Moving Objects

For a moving object with constant force applied, we can derive a simpler expression for power in terms of force and velocity.

**Derivation:** Derive $P = Fv$ for constant force parallel to motion

*Starting from:* Starting from $P = \frac{W}{t}$ and $W = Fd$

1. Substitute work into the power equation: $P = \frac{Fd}{t}$
2. Since $\frac{d}{t} = v$ (constant speed), this simplifies to $P = Fv$

*Conclusion:* This formula gives instantaneous power when $v$ is the instantaneous speed of the object.

> **tip**
>
> If force is at an angle $\theta$ to velocity, the general formula is $P = Fv\cos\theta$. If force is perpendicular to velocity, $\cos 90^\circ = 0$, so power is zero.

**Worked example:** A cyclist travels at constant speed 8 m/s against a total resistive force of 30 N. What power must the cyclist produce to maintain this speed?

1. At constant speed, driving force equals resistive force, so $F = 30 \text{ N}$
2. Use $P = Fv$:
3. $$P = 30 \times 8 = 240 \text{ W}$$

## Electrical Power

**Electrical Power** — Rate of electrical energy transfer in a component, calculated from potential difference and current.

- Base formula (always valid): $P = VI$
- For ohmic components (constant $R$): $P = I^2 R = \frac{V^2}{R}$
- Where $V$ = potential difference, $I$ = current, $R$ = resistance

**Worked example:** A 5 Ω resistor is connected to a 10 V battery. Calculate power dissipated in the resistor.

1. Use $P = \frac{V^2}{R}$:
2. $$P = \frac{10^2}{5} = \frac{100}{5} = 20 \text{ W}$$
3. Check with $P = VI$: $I = V/R = 2 A$, so $P = 10 \times 2 = 20 W$, which matches.

## Power and Efficiency

No real energy conversion process is 100% efficient. Some energy is always lost as heat to the surroundings, so useful output power is always less than total input power.

**Efficiency** — Efficiency is the ratio of useful output power to total input power, expressed as a decimal or percentage.

*Notation:* $\eta$

$$\eta = \frac{P_{out}}{P_{in}} \times 100\%$$

**Worked example:** An electric motor has an input power of 2 kW. It lifts a 75 kg mass at a constant speed of 1.5 m/s. Calculate the efficiency of the motor.

1. Calculate useful output power: $F = mg = 75 \times 9.81 = 735.75 N$
2. Use $P_{out} = Fv = 735.75 \times 1.5 = 1103.6 W$
3. Convert input power to watts: $P_{in} = 2 kW = 2000 W$
4. Calculate efficiency:
5. $$\eta = \frac{1103.6}{2000} \times 100\% = 55.2\% \approx 55\%$$

> **warning**
>
> Efficiency can never be greater than 1 (100%). If you get a value over 100%, you swapped input and output power.

## Common pitfalls

- **Wrong:** Confusing energy and power, using total energy instead of rate of energy transfer
  - Why it fails: Power is a rate, not a total quantity. Mixing them up leads to wrong units and wrong values.
  - Correct: Always check units first: power must be in watts (J/s), energy in joules. Divide total energy by time to get power.
- **Wrong:** Forgetting to convert kW to W (or other unit prefixes) before calculation
  - Why it fails: CIE examiners regularly test unit conversion by giving power in kilowatts, leading to answers off by a factor of 1000.
  - Correct: Always convert all quantities to base SI units (watts, seconds, newtons) before starting any calculation.
- **Wrong:** Using $P=Fv$ when force and velocity are perpendicular
  - Why it fails: The formula $P=Fv$ assumes force is parallel to motion. No work is done when force is perpendicular, so power is zero.
  - Correct: Use $P = Fv\cos\theta$, where $\theta$ is the angle between force and velocity.
- **Wrong:** Swapping input and output power in the efficiency formula
  - Why it fails: Misremembering the order of the ratio leads to efficiency greater than 100%, which is impossible.
  - Correct: Efficiency = useful output / total input, always. Output can never be larger than input.
- **Wrong:** Using $P=\frac{V^2}{R}$ for non-ohmic components like diodes
  - Why it fails: Resistance is not constant for non-ohmic components, so the derived formula does not hold.
  - Correct: Always use $P=VI$ for any electrical component, this formula is always valid.

## Cheatsheet

| Quantity | Formula | SI Units |
| --- | --- | --- |
| Average Power | $P = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$ | watts (W = J/s) |
| Instantaneous Mechanical Power | $P = Fv\cos\theta$ | watts |
| General Electrical Power | $P = VI$ | watts |
| Electrical Power (ohmic) | $P = I^2 R = \frac{V^2}{R}$ | watts |
| Efficiency | $\eta = \frac{P_{out}}{P_{in}} \times 100\%$ | unitless / % |

## What's next

Power is a core concept that appears across multiple topics in CIE A-Level Physics. It is used to analyse the motion of vehicles, calculate energy losses in electrical circuits, and understand energy generation and conservation. Mastery of power calculations is essential for scoring full marks on both multiple choice and extended response questions, as it is often combined with other concepts like forces, energy conservation and Ohm's law. Building a strong understanding of power now will make more advanced topics easier to tackle later.

- [Conservation of Energy](https://www.owlsprep.com/study/cie-9702-u5-conservation-of-energy/)
- [Energy efficiency](https://www.owlsprep.com/study/cie-9702-u5-energy-efficiency/)
- [Deformation of solids](https://www.owlsprep.com/study/cie-9702-u6-overview/)

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