# Estimation of physical quantities

> CIE A-Level Physics · 9702
> Source: https://www.owlsprep.com/study/cie-9702-u1-estimation-of-physical-quantities/

This module teaches you to produce reasonable order-of-magnitude estimates for physical quantities, a core skill regularly tested in CIE Paper 1 multiple-choice questions. You will learn strategies for both memorised and derived estimates.

**Prerequisites:** [SI units and scientific notation](https://www.owlsprep.com/study/cie-9702-u1-si-units/)

## Learning objectives

- Estimate values of common physical quantities to appropriate orders of magnitude
- Identify unreasonable estimates for given physical contexts
- Combine known estimates to find approximate values for derived quantities

## Orders of magnitude and estimation basics

**Order of magnitude estimation** — A rough approximation of a quantity's size, rounded to the nearest power of 10, used when exact values are not required

*Notation:* Written as $\sim 10^n$

*Example:* The height of an adult human is ~ 2 m, so its order of magnitude is $10^0$ m

For estimation questions in CIE exams, you only need to get the power of 10 correct. Factors of 2 or 3 do not change the order of magnitude, so you can round freely during calculations.

**Worked example:** Estimate the order of magnitude of the number of breaths an adult takes in one year.

1. Start with an estimate for breathing rate: typical resting rate is ~ 12 breaths per minute, which is ~ 0.2 breaths per second.
2. Calculate the number of seconds in one year:
3. $$60 \times 60 \times 24 \times 365 \approx 3 \times 10^7$$
4. Multiply rate by time to get total breaths: $0.2 \times 3 \times 10^7 = 6 \times 10^6$.
5. Round to nearest power of 10: $6 \times 10^6$ is closer to $10^7$ than $10^6$, so the order of magnitude is $10^7$ breaths per year.

## Common memorised estimates

CIE exams expect you to have memorised reasonable estimates for a range of common everyday and atomic-scale quantities. Most multiple-choice estimation questions test your knowledge of these standard values.

> **warning**
>
> Always check the units of the options given. Questions often mix grams and kilograms, or nanometers and meters, to catch candidates who forget to convert units.

**Worked example:** What is the best estimate for the diameter of an atom?

1. Recall the standard estimate for atomic diameter: the size of a small atom like hydrogen is roughly 0.1 nanometers.
2. Convert to SI units (meters): 0.1 nm = $0.1 \times 10^{-9}$ m = $1 \times 10^{-10}$ m.
3. The order of magnitude is $10^{-10}$ m, which is the correct answer.

## Estimating derived quantities

For quantities you haven't memorised, break the quantity down into simpler terms that you can estimate, then combine using the relevant physical formula. You only need to keep track of powers of 10.

**Worked example:** Estimate the order of magnitude of the mass of air in a small classroom.

1. Density of air $\rho = \frac{m}{V}$, so mass $m = \rho V$. We need estimates for density of air and volume of the classroom.
2. Density of air is approximately $1$ kg m$^{-3}$, so order of magnitude $10^0$ kg m$^{-3}$.
3. Estimate classroom dimensions: 4 m × 5 m × 3 m = 60 m³ ≈ 10² m³.
4. Multiply: $m = 1 \times 60 = 60$ kg, which rounds to order of magnitude $10^2$ kg.

> **tip**
>
> If your final value is 5 × 10ⁿ or higher, round up to 10ⁿ⁺¹. Values lower than 5 × 10ⁿ round down to 10ⁿ.

## Common pitfalls

- **Wrong:** Forgetting to convert all quantities to the same units before estimating
  - Why it fails: Exam questions intentionally use mixed units in options to test unit awareness, leading to incorrect order of magnitude
  - Correct: Always convert all values to SI units before calculating your estimate
- **Wrong:** Wasting time trying to calculate an exact value instead of rounding early
  - Why it fails: Estimation questions only require the correct power of 10, exact values are never needed
  - Correct: Round all intermediate values to the nearest power of 10 early in your calculation to save time
- **Wrong:** Mixing up the diameter of an atom and an atomic nucleus
  - Why it fails: A common mistake is to use 10⁻¹⁵ m for an atom or 10⁻¹⁰ m for a nucleus, which is five orders of magnitude wrong
  - Correct: Memorise: atom ~ 10⁻¹⁰ m, nucleus ~ 10⁻¹⁵ m
- **Wrong:** Incorrectly rounding 70 kg (mass of adult) to order 10¹ kg
  - Why it fails: 70 kg = 7 × 10¹ kg, which is closer to 10 × 10¹ = 10² kg than 1 × 10¹ kg
  - Correct: Always round the coefficient first before writing the final order of magnitude

## Cheatsheet

| Quantity | Typical order of magnitude (SI units) |
| --- | --- |
| Height of adult human | $10^0$ m |
| Mass of adult human | $10^2$ kg |
| Mass of an apple | $10^{-1}$ kg |
| Mass of a car | $10^3$ kg |
| Speed of sound in air | $10^2$ m/s |
| Motorway car speed | $10^1$ m/s |
| Diameter of atom | $10^{-10}$ m |
| Diameter of atomic nucleus | $10^{-15}$ m |
| Wavelength of visible light | $10^{-6}$ m |
| Mass of an electron | $10^{-30}$ kg |
| Density of air | $10^0$ kg m$^{-3}$ |

## What's next

Estimation is a foundational skill that you will use throughout your A-Level Physics course, from checking that calculation answers are physically reasonable to planning practical experiments. It is one of the easiest topics to pick up marks for in Paper 1, so investing time to memorise the common values pays off in early exams. Estimation also links closely to unit analysis and dimensional checking, which help you verify your answers in longer calculation questions.

- [Measurement uncertainties](https://www.owlsprep.com/study/cie-9702-u1-measurement-uncertainties/)
- [Kinematics](https://www.owlsprep.com/study/cie-9702-u2-overview/)
- [Distance, displacement, speed, velocity and acceleration](https://www.owlsprep.com/study/cie-9702-u2-distance-displacement-speed-velocity-and/)

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