Estimation and Orders of Magnitude

One of the most valuable — and least drilled — skills in physics is the ability to make a rough-but-reliable estimate of a physical quantity. A good physicist can tell you, to within a factor of ten, the kinetic energy of a running human, the number of atoms in a teaspoon of water, or the frequency of a typical human heartbeat, without looking anything up.

OCR examines this explicitly in Module 2 of the A-Level Physics A specification, under "Estimation of physical quantities". You should be able to: give an order-of-magnitude estimate; use approximate values for common constants; sanity-check a numerical answer; and perform simple dimensional analysis. This lesson trains all four skills.

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What Is an Order of Magnitude?

The order of magnitude of a quantity is the power of ten closest to it. More formally, if a quantity is written as a × 10ⁿ, with 1 ≤ a < 10, then the order of magnitude is n. (Some textbooks round differently — OCR accepts either convention as long as it is consistent.)

For example:

Quantity	Value	Order of Magnitude
Diameter of a hydrogen atom	1 × 10⁻¹⁰ m	−10
Thickness of a sheet of paper	1 × 10⁻⁴ m	−4
Height of a human	1.7 m	0
Height of Mount Everest	8.8 × 10³ m	4
Distance to the Moon	3.8 × 10⁸ m	8
Distance to the nearest star	4 × 10¹⁶ m	16

When two quantities differ by more than an order of magnitude, one is clearly much bigger than the other. When they differ by less than an order, they are in the same "ballpark".

Exam Tip: If OCR asks you to "estimate the order of magnitude" of a quantity, you only need to give the power of ten. A more exact figure is not required and can lose marks for over-precision on an estimation question.

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Some Quantities Every A-Level Physicist Should Know

You should know the approximate values of the following, to within an order of magnitude, without a data sheet.

Quantity	Typical Value
Mass of an electron	10⁻³⁰ kg
Mass of a proton	10⁻²⁷ kg
Mass of a human	70 kg
Mass of a car	10³ kg
Mass of the Earth	6 × 10²⁴ kg
Radius of an atom	10⁻¹⁰ m
Radius of a nucleus	10⁻¹⁵ m
Radius of the Earth	6 × 10⁶ m
Radius of the Sun	7 × 10⁸ m
Speed of sound in air	340 m s⁻¹
Speed of light in vacuum	3 × 10⁸ m s⁻¹
Acceleration of free fall (UK)	9.81 m s⁻²
Atmospheric pressure	10⁵ Pa
Density of water	10³ kg m⁻³
Age of the universe	4 × 10¹⁷ s (≈ 14 × 10⁹ years)
Avogadro constant	6 × 10²³ mol⁻¹
Boltzmann constant	1.4 × 10⁻²³ J K⁻¹
Planck constant	6.6 × 10⁻³⁴ J s
Elementary charge	1.6 × 10⁻¹⁹ C

You are not expected to recite the exact value of Planck's constant — the data sheet provides it — but you should be able to say "about 10⁻³⁴ J s" and use that in an estimation question.

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Fermi Estimation

The Italian-American physicist Enrico Fermi was legendary for his ability to make good estimates with minimal information. A famous example: on the morning of the Trinity nuclear test in 1945, Fermi stood at the observation point, dropped small pieces of paper, and by watching how far the blast wind blew them, estimated the explosive yield of the world's first atomic bomb to within a factor of two of the true value — before any instruments had reported in.

The trick Fermi used is called Fermi estimation: break a hard question into smaller, easier sub-questions, each of which you can estimate to within a factor of 10, and combine the results. Errors tend to cancel, so your final answer is usually surprisingly accurate.

Worked Example 1 — A Classic Fermi Question

Estimate the number of breaths a typical human takes in a lifetime.

Solution:

• Breaths per minute: ≈ 15 (a reasonable resting rate)
• Minutes per hour: 60
• Hours per day: 24
• Days per year: 365 (≈ 400 for round numbers)
• Human lifespan: ≈ 80 years

Multiply:

15 × 60 × 24 × 400 × 80 ≈ 15 × 60 × 24 × 3.2 × 10⁴ ≈ 900 × 24 × 3.2 × 10⁴ ≈ 2.2 × 10⁴ × 3.2 × 10⁴ ≈ 7 × 10⁸ breaths

About a billion — the order of magnitude is 10⁸ to 10⁹. An exam question would accept anything in that range.

Worked Example 2 — Estimating Energy of a Falling Apple

An apple falls from a tree of height 3 m. Estimate the kinetic energy with which it hits the ground.

Solution:

• Mass of an apple: ≈ 0.2 kg
• Height: 3 m
• g: ≈ 10 m s⁻² (round for estimation)

Gravitational PE lost = kinetic energy gained:

E_k ≈ mgh = 0.2 × 10 × 3 = 6 J

Order of magnitude: 10⁰ to 10¹ J.

Worked Example 3 — Rough Number of Atoms in a Human

Estimate the number of atoms in a 70 kg human.

Solution: