Physics › Measurements › Estimation and orders of magnitude
Estimation and orders of magnitude
An order-of-magnitude estimate is a rough answer worked in powers of ten, built from a few sensible assumptions in a few seconds. Its job is to show whether a full calculation has come out at a plausible size.
Pick your board and the few notes written for the other boards quietly fold away, here and in the practice players. Nothing is deleted: every folded piece reopens on a tap.
Watch Estimation and orders of magnitude on the InkPhysics YouTube channel
Builds on SI units and prefixes.
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 audit what your calculator says.
COMMON MISCONCEPTION
If it came out of the calculator, it must be right.
The unit of the game is the power of ten
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 are seventeen rungs apart on the ladder, a factor of , and you can say so without ever 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.
Comparison beats recall from nowhere, every time. A grand piano is unfamiliar; eight Labradors is not. Anchor to something you can picture and the estimate lands within an order of magnitude, which is all the game asks of you.
Estimate, then derive
What is really being tested is the derived estimate, where rough inputs get fed through real physics. Try the energy to climb a flight of stairs. Lifting a 70 kg person needs a force near 700 N, taking g as 10, and a flight rises about 3 m, so the energy is roughly 700 × 3 ≈ 2000 J, order of magnitude J. Round brutally as you go. The powers of ten do the work, and the details cancel out of an order-of-magnitude answer anyway.
Then let the estimate do its real job. Run it before the precise calculation, and compare. If the calculator says J for the stairs, the estimate has just caught a slipped unit or a mistyped power, three orders before the examiner does.
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. A few billion seconds each: spend them well.
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. Stand on one heel and the same force concentrates tenfold. Hence the wince.
ASSESSMENT FOCUS
- Asked for an order of magnitude, answer with a power of ten and a unit. A four-figure calculation is wasted time here, and it can bury the mark you were after.
- State the assumptions you feed in. “Taking the mass of an adult as 70 kg” is creditable working, not padding, and 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, and that is fine. Anyone whose calculation of this quantity produces J has found an error, not a big staircase.
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 from GCSE.
- Significant figures, units and checking is the maths this lesson leans on, worked through from GCSE.
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 audit what your calculator says.
Open the full revision checklist to track your progress across the whole unit.