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How can orders of magnitude and sensible estimates let a physicist sanity-check an answer without precise data?

Orders of magnitude, estimation of approximate values of physical quantities to the nearest power of ten, and using such estimates to check the plausibility of calculated results.

A focused answer to AQA A-Level Physics 3.1.3, covering orders of magnitude, estimating physical quantities to the nearest power of ten, and using these estimates to check whether a calculated result is plausible.

Generated by Claude Opus 4.88 min answer

Reviewed by: AI editorial process; not yet individually human-reviewed

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  1. What this dot point is asking
  2. Orders of magnitude
  3. Useful reference values
  4. Making an estimate
  5. Using estimates to check answers
  6. Try this

What this dot point is asking

AQA specification point 3.1.3 wants you to estimate physical quantities to the nearest order of magnitude, express results as powers of ten, and use estimates to judge whether a more detailed calculation is plausible.

Orders of magnitude

Comparing two quantities by their orders of magnitude tells you roughly how many times bigger one is than the other. Two quantities that differ by three orders of magnitude differ by a factor of about 103=100010^3 = 1000. Order-of-magnitude thinking is how physicists handle quantities that span the whole range of nature, from the radius of a proton (10−15 m10^{-15} \text{ m}) to the size of the observable universe (1026 m10^{26} \text{ m}). To find the order of magnitude of a number, write it in standard form a×10na \times 10^n with 1≤a<101 \le a < 10; the order of magnitude is 10n10^n if a<5a < 5 and 10n+110^{n+1} if a≥5a \ge 5. For example, 4.2×1074.2 \times 10^7 is of order 10710^7, while 7.8×1077.8 \times 10^7 is of order 10810^8.

Useful reference values

Knowing a few benchmark values makes estimation reliable:

  • Size of an atom: about 10−10 m10^{-10} \text{ m}.
  • Size of a nucleus: about 10−15 m10^{-15} \text{ m}.
  • Mass of a person: about 102 kg10^2 \text{ kg}.
  • Speed of light: about 3×108 m s−13 \times 10^8 \text{ m s}^{-1}.
  • Acceleration of free fall near Earth: about 10 m s−210 \text{ m s}^{-2}.
  • Atmospheric pressure: about 105 Pa10^5 \text{ Pa}.
  • Avogadro constant: about 6×1023 mol−16 \times 10^{23} \text{ mol}^{-1}.

Making an estimate

The Fermi method breaks a hard quantity into easy-to-guess factors. Round each to one significant figure (or one power of ten), then combine.

Using estimates to check answers

Try this

Q1. State the order of magnitude of the ratio of the size of an atom to the size of its nucleus. [1 mark]

  • Cue. 10−1010−15=105\dfrac{10^{-10}}{10^{-15}} = 10^5, so about five orders of magnitude.

Q2. Estimate the number of grains of sand that would fill a one-litre bottle, stating your assumptions. [3 marks]

  • Cue. Assume a grain is about 0.5 mm0.5 \text{ mm} across, so its volume is about 10−10 m310^{-10} \text{ m}^3; one litre is 10−3 m310^{-3} \text{ m}^3, giving of order 10710^7 grains.

Q3. State the order of magnitude of atmospheric pressure in pascals. [1 mark]

  • Cue. About 105 Pa10^5 \text{ Pa}.

Exam-style practice questions

Practice questions written in the style of AQA exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AQA 20183 marksEstimate, to the nearest order of magnitude, the number of water molecules in a 1.0 cm31.0 \text{ cm}^3 drop of water. State the assumptions you make. The molar mass of water is 18 g mol−118 \text{ g mol}^{-1} and NA=6.0×1023 mol−1N_A = 6.0 \times 10^{23} \text{ mol}^{-1}.
Show worked answer →

Assume the density of water is 1.0 g cm−31.0 \text{ g cm}^{-3}, so a 1.0 cm31.0 \text{ cm}^3 drop has a mass of about 1.0 g1.0 \text{ g}.

The number of moles is 1.018≈0.056 mol\dfrac{1.0}{18} \approx 0.056 \text{ mol}.

The number of molecules is 0.056×(6.0×1023)≈3×10220.056 \times (6.0 \times 10^{23}) \approx 3 \times 10^{22}, which to the nearest order of magnitude is 102210^{22} molecules.

Markers reward stating the density assumption, finding the moles, multiplying by the Avogadro constant, and giving the order of magnitude.

AQA 20213 marksA student calculates the power output of a domestic wind turbine to be 2×109 W2 \times 10^9 \text{ W}. Use an order-of-magnitude estimate to explain why this answer is implausible, and suggest what the student may have done wrong.
Show worked answer →

A domestic wind turbine has a sensible power output of roughly a few kilowatts, of order 103 W10^3 \text{ W}, comparable to household demand. The student's answer of 2×109 W2 \times 10^9 \text{ W} is about a million times too large, comparable to a small power station.

Such a discrepancy of around six orders of magnitude points to a unit or power-of-ten error, for example failing to convert kilometres to metres, mixing up a prefix (mega for kilo), or an arithmetic slip with powers of ten.

Markers reward a sensible benchmark (kilowatts), recognising the answer is far too large, and identifying a likely unit or power-of-ten error.

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