Physics › Measurements › Estimation and orders of magnitude
Estimation and orders of magnitude
An order-of-magnitude estimate gives a quantity to the nearest power of ten, worked from a few benchmark values and rounded arithmetic. Physicists use it to test whether a full calculation has returned a plausible size, and it quickly catches a calculator slip or a unit left unconverted.
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Builds on SI base and derived units.
IN THIS TOPIC
- Give a quantity's order of magnitude as the nearest power of ten.
- Carry a handful of benchmarks, chain them through an equation, and use the result to check a calculated value.
COMMON MISCONCEPTION
If it came out of the calculator, it must be right.
A calculator will not detect an incorrect input. Estimate the order of magnitude first so that you can identify a misplaced power of ten.
Orders of magnitude
An order of magnitude is a power of ten. To give one, round the quantity to the nearest power. 700 J sits nearer than , so its order of magnitude is J. Two quantities differ by three orders of magnitude when one is about a thousand times the other.
Working in exponents makes enormous comparisons manageable. An atom at m and the Earth at m differ by seventeen orders of magnitude, a factor of , which can be stated without writing the zeros out.
Carry benchmarks
Estimation runs on a pocketful of memorised anchors. A door is about 2 m; a person is around 70 kg; water has a density of 1000 kg m−3 and air roughly 1.2 kg m−3; a car travels near 30 m s−1 on a motorway; mains appliances draw a few kilowatts.
Use familiar reference values, such as 70 kg for an adult, and state each assumption. An estimate only needs to establish the correct order of magnitude.
Estimate, then derive
Choose plausible benchmark values, state each assumption and combine them with the relevant equation. Give the final estimate as the nearest power of ten with a unit. Compare it with the full calculation to identify errors in units, prefixes or powers of ten.
For the energy needed to climb a flight of stairs: lifting a 70 kg person needs a force near 700 N, taking g as 10 m s−2, and a flight rises about 3 m, so the energy is roughly 700 × 3 ≈ 2000 J, order of magnitude J. Rounding each input to one significant figure does not change the power of ten.
Make the estimate before the precise calculation, then compare the two. A calculator result of J for the stairs differs from the estimate by three orders of magnitude, which indicates a unit slip or a mistyped power.
WORKED EXAMPLE
A sprinter's kinetic energy
Estimate the kinetic energy of a person sprinting flat out.
Pick benchmarks: a person is about 70 kg, and a fast sprint is about 8 m s−1.
E = ½mv2 = ½ × 70 × 82 = 2240 J, so the order of magnitude is 103 J.
The benchmarks could each be off by tens of per cent without moving the power of ten, which is what makes the method dependable.
GUIDED PRACTICE
Seconds in a lifetime
Estimate, to the nearest order of magnitude, the number of seconds in a human lifetime. Set out the chain of factors.
Show the working
About 80 years × 365 days × 24 hours × 3600 seconds ≈ 2.5 × 109.
Order of magnitude: 109 s.
INDEPENDENT PRACTICE
The pressure under your feet
Estimate the pressure a standing person exerts on the floor, choosing your own benchmarks for weight and foot area.
Show the working
Weight about 700 N; two feet cover roughly 0.03 m2.
p = F/A ≈ 700/0.03 ≈ 2 × 104 Pa, so 104 Pa, roughly a quarter of atmospheric pressure. Standing on one heel reduces the contact area by about a factor of ten, so the pressure rises by about the same factor.
ASSESSMENT FOCUS
- Asked for an order of magnitude, answer with a power of ten and a unit. A four-figure calculation is not what the question asks for.
- State the assumptions you feed in. “Taking the mass of an adult as 70 kg” is creditable working. Estimates want one significant figure, with g ≈ 10 throughout.
- An answer three orders from your estimate is almost always a unit slip, grams for kilograms or millimetres for metres. Check the conversions before you check the physics.
CHECK YOURSELF
Estimate, to the nearest order of magnitude, the energy needed to climb one flight of stairs.
Show a hint
The force is your weight. Take g as 10 and a flight as about 3 m.
Show the answer
Weight of a person: about 70 kg × 10 = 700 N. Height of a flight: about 3 m.
Energy = force × distance ≈ 700 × 3 = 2100 J, so the order of magnitude is J, a couple of kilojoules.
The exact answer depends on the person and the staircase, so only the power of ten is required. A calculated value of J for this quantity indicates an error in units or in a power of ten.
Estimate before you calculate.
The calculator's answer must land near it.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their mark schemes on the estimation and orders of magnitude questions page.
WHERE TO GO NEXT
- Standard form and prefixes is the maths this lesson leans on, worked through step by step.
- Significant figures, units and checking is the maths this lesson leans on, worked through step by step.
CHECK YOUR PROGRESS
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- Give a quantity's order of magnitude as the nearest power of ten.
- Carry a handful of benchmarks, chain them through an equation, and use the result to check a calculated value.
Open the full revision checklist to track your progress across the whole unit.