Physics › Measurements › Estimation and orders of magnitude
Estimation and orders of magnitude
Physicists check a calculation the way you check change from a shop: a rough figure first, held ready before the exact one arrives. The rough figure lives in powers of ten, and building it takes seconds.
Builds on SI units and prefixes.
IN THIS TOPIC
- State the order of magnitude of a quantity as the nearest power of ten.
- Carry a small set of benchmark values and scale unfamiliar quantities against them.
- Chain estimates through an equation to produce a derived estimate, and use it to audit a calculated answer.
WHAT YOU PROBABLY THINK
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 a quantity's order of magnitude, round it to the nearest power: 700 J sits nearer 103 than 102, so its order of magnitude is 103 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 10−10 m and the Earth at 107 m are seventeen rungs apart on the ladder, a factor of 1017, 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.
The trick is always comparison, never recall from nowhere. A grand piano is unfamiliar; eight Labradors is not. Anchoring to something you can picture keeps an estimate within an order of magnitude, which is all the game requires.
Estimate, then derive
The specification asks for derived estimates: feed rough inputs through real physics. Estimate 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 103 J. Round brutally as you go; the powers of ten do the work and the details cancel out of an order-of-magnitude answer.
Then let the estimate do its real job. Run it before the precise calculation, and compare. If the calculator says 106 J for the stairs, the estimate has just caught a slipped unit or a mistyped power, three orders before the examiner does.
THE EXAM BIT
- When a question asks for an order of magnitude, the answer is a power of ten, with a unit. A full four-figure calculation is wasted time and can even obscure the mark.
- Estimates want one significant figure and g ≈ 10. Nobody is testing your recall of 9.81 here.
- State the assumptions you feed in: “taking the mass of an adult as 70 kg” is part of the creditable working.
- Derived estimates earn method marks. A sensible chain with rough inputs beats a blank space every time.
- An answer three orders of magnitude from your estimate almost always means a unit slip: grams for kilograms, or millimetres for metres. Check the conversions before anything else.
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 103 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 106 J has found an error, not a big staircase.
Estimate before you calculate.
The calculator's answer must land near it.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.