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3Working as a PhysicistFoundation

Estimation and Orders of Magnitude

How much energy does a sprinter have at full speed? How many air molecules are in this room? Physicists answer questions like these in seconds, without looking anything up. Estimation\textbf{Estimation} means choosing sensible values and getting within a factor of ten of the true answer.

25 min Topic 1 · Working as a Physicist

What you'll be able to do

  • Recall approximate values of common physical quantities
  • Make reasoned estimates and state the assumptions behind them
  • Express answers to the nearest order of magnitude
  • Use estimates to check whether calculated answers are reasonable
1

Orders of magnitude

The order of magnitude\textbf{order of magnitude} of a quantity is the power of ten closest to it. A 70 kg person has mass of order 10210^{2} kg; the Earth–Sun distance, 1.5×10111.5\times10^{11} m, is of order 101110^{11} m.

A value such as 4×1054\times10^{5} is usually rounded to 10510^{5} or 10610^{6} by comparing it with 10≈3.2\sqrt{10} \approx 3.2: below that, round down; above it, round up.

Two quantities are "three orders of magnitude apart" if their ratio is about 10310^{3}.

Example 1By how many orders of magnitude does the diameter of an atom (∼10−10\sim10^{-10} m) exceed that of a nucleus (∼10−15\sim10^{-15} m)?
1Ratio =10−10/10−15=105= 10^{-10}/10^{-15} = 10^{5}.
AnswerFive orders of magnitude.
2

Values worth knowing

Masses\textbf{Masses}: apple 0.10.1 kg; adult 7070 kg; car 10001000 kg; bus 10410^{4} kg.

Lengths\textbf{Lengths}: atom 10−1010^{-10} m; human hair width 10−410^{-4} m; height of a person 1.71.7 m; height of a room 2.52.5 m; Earth’s radius 6×1066\times10^{6} m.

Speeds\textbf{Speeds}: walking 1.5 m s−11.5\ \mathrm{m\,s^{-1}}; sprinter 10 m s−110\ \mathrm{m\,s^{-1}}; motorway car 30 m s−130\ \mathrm{m\,s^{-1}}; sound in air 340 m s−1340\ \mathrm{m\,s^{-1}}; light 3×108 m s−13\times10^{8}\ \mathrm{m\,s^{-1}}.

Energy and power\textbf{Energy and power}: kettle 22 kW; light bulb 1010–6060 W; human resting power 100100 W; food energy per day 10710^{7} J.

Tip — Choose values you could defend in a sentence. Examiners accept a range of sensible values; they penalise answers built on absurd ones.

3

Making an estimate

Break the problem into quantities you can estimate, and write down each assumption — this is where the marks are.

Use rounded values, often to one significant figure, and keep the arithmetic simple.

Give the final answer to one significant figure or as an order of magnitude. Quoting three significant figures from rough inputs implies precision you do not have.

Example 1Estimate the kinetic energy of a sprinter at top speed.
1Assume mass ≈70\approx 70 kg and top speed ≈10 m s−1\approx 10\ \mathrm{m\,s^{-1}}.
2Ek=12×70×102=3500E_{k} = \tfrac12 \times 70 \times 10^{2} = 3500 J.
AnswerAbout 4×1034\times10^{3} J — of order 10310^{3} J.
Example 2Estimate the mass of air in a classroom. The density of air is about 1.2 kg m−31.2\ \mathrm{kg\,m^{-3}}.
1Assume the room is 10×8×310 \times 8 \times 3 m, volume 240 m3240\ \mathrm{m^{3}}.
2m=ρV=1.2×240≈290m = \rho V = 1.2 \times 240 \approx 290 kg.
AnswerAbout 300300 kg — of order 10210^{2} kg.
4

Estimates as a check

Before trusting a calculated answer, estimate what it should be. A factor of 10310^{3} discrepancy usually means a missed prefix, such as grams for kilograms or millimetres for metres.

Watch the powers of ten in particular: a missed 10−610^{-6} from mm2\mathrm{mm^{2}} or μ\muF turns a correct method into an answer that is wildly off.

In practical work, estimate expected values before measuring, so you notice immediately if equipment is set up wrongly.

Tip — In a "show that" question, the value given is a built-in estimate. If your working lands far from it, find the error before moving on.

Equation recap

10≈3.2\sqrt{10} \approx 3.2
Rounding boundary for orders of magnitude.
m=ρVm = \rho V
Common estimation route to mass.

Common mistakes to avoid

Giving an estimate to four significant figures.
Rough inputs justify one significant figure at most.
Not stating assumed values.
The assumptions are usually what earns the marks.
Using unrealistic values, such as a 10 kg adult.
Use values you could justify, such as 70 kg.
Ignoring an answer that is 10310^{3} times too large.
Check for a missed prefix conversion.

Key takeaways

  • An order of magnitude is the nearest power of ten.
  • Know approximate masses, lengths, speeds and powers of everyday things.
  • State every assumption when estimating.
  • Give estimates to one significant figure or an order of magnitude.
  • Use estimates to check calculated answers for prefix errors.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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