# Energy, work and power

> CIE A-Level Mathematics · 9709 Mechanics Unit 3
> Source: https://www.owlsprep.com/study/cie-9709-u3-energy-work-and-power/

This sub-topic introduces core definitions for work done, kinetic energy, gravitational potential energy and power, and shows how to apply conservation of mechanical energy to solve practical Mechanics motion problems.

**Prerequisites:** [Forces and Newton's laws of motion](https://www.owlsprep.com/study/cie-9709-u3-forces-newtons-laws/); [Kinematics of straight line motion](https://www.owlsprep.com/study/cie-9709-u3-kinematics-straight-line/)

## Learning objectives

- Define work done by a constant force and calculate it for different force directions
- Distinguish between kinetic energy and gravitational potential energy
- Apply conservation of mechanical energy to solve motion problems
- Calculate power and efficiency for mechanical systems

## How often is this tested?

Based on OwlsPrep's analysis of official CIE 9709 M1 past papers (2016–2025): **Energy, work and power** appears **125 times** in the last 10 years — **28.2%** of all M1 questions (125 of 443).

Most-tested forms: Power (63), Conservation of energy (56), Kinetic energy (14), Work done by a force (13), Gravitational potential energy (8).

## Work Done by a Constant Force

**Work done by a constant force** — The energy transferred when a force moves an object through a displacement. \theta is the angle between the force and displacement vectors.

*Notation:* W = F s \cos\theta

*Example:* Work done by a 10 N force pulling an object 2 m at 30° to the horizontal is $10 \times 2 \times \cos 30^\circ \approx 17.3$ J.

Work done can be positive, negative or zero. If the force acts in the same direction as displacement ($\theta = 0^\circ$), $\cos\theta = 1$ so $W = Fs$, meaning positive work (energy is added to the object). If the force opposes motion ($\theta = 180^\circ$), work is negative, meaning energy is removed from the object. If the force is perpendicular to displacement ($\theta = 90^\circ$), work done is zero.

**Worked example:** A 5 kg block is pulled 4 m up a rough slope inclined at 20° to the horizontal by a rope parallel to the slope. Friction has magnitude 8 N, and the block moves at constant speed. Calculate the total work done by all forces acting on the block.

1. List all forces parallel to displacement (along the slope): weight component is down the slope:
2. $$mg \sin\theta = 5 \times 9.8 \times \sin 20^\circ \approx 16.8 \text{ N}$$
3. Tension up the slope balances the sum of friction and weight component for constant speed:
4. $$T = 16.8 + 8 = 24.8 \text{ N}$$
5. Calculate work done by each force: tension is parallel to displacement:
6. $$W_T = 24.8 \times 4 = 99.2 \text{ J}$$
7. Friction and weight component oppose displacement, so their work is negative:
8. $$W_f + W_g = (-8 \times 4) + (-16.8 \times 4) = -99.2 \text{ J}$$
9. Total work done is the sum of all individual work values:
10. $$W_{\text{total}} = 99.2 - 99.2 = 0 \text{ J}$$
11. This matches the work-energy principle: no change in kinetic energy means total work done is zero.

> **Exam tip:** Always check the angle between the force and displacement; many candidates incorrectly use $\sin\theta$ instead of $\cos\theta$.

## Kinetic and Gravitational Potential Energy

**Kinetic Energy (KE)** — The energy an object possesses due to its motion, measured in joules (J)

*Notation:* KE = \frac{1}{2}mv^2

*Example:* A 2 kg object moving at 3 m s⁻¹ has $KE = 0.5 \times 2 \times 3^2 = 9$ J.

**Gravitational Potential Energy (GPE)** — The change in energy of an object when it changes height by $\Delta h$ in a uniform gravitational field

*Notation:* \Delta PE = mg\Delta h

*Example:* Lifting a 10 kg mass by 2 m increases GPE by $10 \times 9.8 \times 2 = 196$ J.

The work-energy principle states that the total work done by all forces acting on an object equals the change in its kinetic energy: $W_{\text{total}} = \Delta KE$. This principle holds whether or not there are resistive forces, making it very flexible for problem solving.

**Worked example:** A car of mass 1200 kg is moving at 15 m s⁻¹. The brakes apply a constant braking force of 4500 N. Find the distance the car travels before stopping.

1. Calculate initial kinetic energy of the car:
2. $$KE_{\text{initial}} = \frac{1}{2} \times 1200 \times 15^2 = 135000 \text{ J}$$
3. Final KE is 0 when the car stops, so change in KE is:
4. $$\Delta KE = 0 - 135000 = -135000 \text{ J}$$
5. Work done by braking force $F$ over distance $d$ is $W = -Fd$. By work-energy principle: $W = \Delta KE$
6. $$-4500d = -135000$$
7. Solve for stopping distance $d$:
8. $$d = \frac{135000}{4500} = 30 \text{ m}$$

## Conservation of Mechanical Energy

When no non-conservative forces (like friction or air resistance) do work on a system, the total mechanical energy (sum of kinetic and potential energy) is conserved. This means we can relate energy at the start of motion to energy at the end without needing to calculate acceleration or time.

**Conservation of Mechanical Energy** — In the absence of non-conservative forces, total mechanical energy remains constant: energy is only converted between KE and PE, not created or destroyed.

**Worked example:** A ball of mass 0.5 kg is dropped from rest from a height of 10 m above the ground. Ignoring air resistance, find its speed when it hits the ground.

1. Set GPE = 0 at ground level. Initial energy: ball is at rest so $KE_{\text{initial}} = 0$
2. $$PE_{\text{initial}} = mgh = 0.5 \times 9.8 \times 10 = 49 \text{ J}$$
3. Final energy at ground level: $PE_{\text{final}} = 0$, so all energy is kinetic:
4. $$KE_{\text{final}} = \frac{1}{2}mv^2$$
5. By conservation of energy, total initial energy equals total final energy:
6. $$0 + 49 = \frac{1}{2}(0.5)v^2 + 0$$
7. Solve for speed $v$:
8. $$v^2 = 196 \implies v = 14 \text{ m s}^{-1}$$

> **info**
>
> If friction is present, the work done against friction equals the difference between initial and final total energy: $E_{\text{initial}} = E_{\text{final}} + W_{\text{against friction}}$.

## Power and Efficiency

**Power** — The rate of doing work, measured in watts (W), where 1 W = 1 J s⁻¹. For a constant force $F$ in the direction of motion at speed $v$, power equals the product of force and speed.

*Notation:* P = \frac{W}{t} = Fv

**Efficiency** — A measure of how much input energy is converted to useful output energy. Efficiency is always less than 100% for real systems due to energy losses like heat from friction.

*Notation:* \text{Efficiency} = \frac{\text{useful power output}}{\text{power input}} \times 100\%

**Worked example:** A car engine has a useful power output of 60 kW and moves at constant speed 20 m s⁻¹ along a horizontal road. Calculate the total resistive force acting on the car.

1. Convert power from kilowatts to watts for consistency of units:
2. $$P = 60 \text{ kW} = 60000 \text{ W}$$
3. At constant speed, driving force equals total resistive force by Newton's first law: $F_{driving} = F_{resistive}$
4. Use the instantaneous power formula $P = Fv$:
5. $$F = \frac{P}{v} = \frac{60000}{20} = 3000 \text{ N}$$
6. Therefore total resistive force acting on the car is 3000 N.

## Common pitfalls

- **Wrong:** Using $W = Fs$ for a force at an angle to displacement, ignoring the $\cos\theta$ term
  - Why it fails: Work done depends on the component of force parallel to displacement, not the full force magnitude
  - Correct: Always use $W = Fs \cos\theta$ where $\theta$ is the angle between the force and displacement vectors
- **Wrong:** Taking $\Delta PE$ as negative when an object gains height
  - Why it fails: Gravitational potential energy increases when height increases, so the change should be positive
  - Correct: Use $\Delta PE = mg(h_{final} - h_{initial})$, which is positive when the object moves upwards
- **Wrong:** Applying conservation of mechanical energy without accounting for work done against friction
  - Why it fails: Conservation of total mechanical energy only holds when there are no non-conservative forces doing work
  - Correct: If friction is present, use: $\text{Initial total energy} = \text{Final total energy} + \text{Work done against friction}$
- **Wrong:** Forgetting to convert power from kilowatts to watts before calculation
  - Why it fails: Power must be in watts to get force in newtons, leading to answers 1000x too large or small
  - Correct: Always multiply kilowatts by 1000 to get watts before substituting into formulas
- **Wrong:** Calculating efficiency as input power divided by output power
  - Why it fails: This gives a value greater than 100% for most real systems, which is impossible
  - Correct: Use: $\text{Efficiency} = \frac{\text{Useful output power}}{\text{Total input power}} \times 100\%$

## Cheatsheet

| Quantity | Formula | Key Notes |
| --- | --- | --- |
| Work done (constant force) | $W = Fs \cos\theta$ | $\theta$ = angle between $F$ and $s$ |
| Kinetic Energy | $KE = \frac{1}{2}mv^2$ | Energy due to motion |
| Change in GPE | $\Delta PE = mg\Delta h$ | Positive when height increases |
| Work-Energy Principle | $W_{total} = \Delta KE$ | Works for all forces |
| Conservation of Energy (no resistance) | $KE_1 + PE_1 = KE_2 + PE_2$ | Total mechanical energy constant |
| Average Power | $P = \frac{W}{t}$ | Units: watts (W) = J/s |
| Instantaneous Power | $P = Fv$ | Force parallel to velocity |
| Efficiency | $\frac{P_{out}}{P_{in}} \times 100\%$ | Always < 100% for real systems |

## What's next

Understanding energy, work and power is foundational for all advanced Mechanics topics in A-level Mathematics, including circular motion, momentum, and rigid body equilibrium. Conservation of energy is a particularly powerful problem-solving tool that often simplifies calculations compared to using Newton's laws directly, especially for motion along curved paths where force varies with position. Mastery of this topic also prepares you to tackle combined Mechanics questions that draw on multiple core concepts, which are common in CIE A-level exams.

- [Momentum](https://www.owlsprep.com/study/cie-9709-u3-momentum/)
- [Probability & Statistics](https://www.owlsprep.com/study/cie-9709-u4-overview/)
- [Data Representation](https://www.owlsprep.com/study/cie-9709-u4-data-representation/)

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