Measurement, units, and the errors that destroy spacecraft
Every quantity in physics is a number and a unit. Losing the second half loses everything.
In 1999 NASA's Mars Climate Orbiter was lost on arrival. One team had supplied thrust figures in pound-force seconds; the flight software expected newton-seconds. The numbers were right and the units were not, so the spacecraft approached far too low and was destroyed.
That is the strongest available argument for taking units seriously, and it is why physics teachers are strict about something that looks like bookkeeping.
The seven base units
The SI system defines seven quantities, and every other unit is built from them.
- metre (m) - length
- kilogram (kg) - mass
- second (s) - time
- ampere (A) - electric current
- kelvin (K) - temperature
- mole (mol) - amount of substance
- candela (cd) - luminous intensity
Everything else is a combination. A newton is kg m/s squared. A joule is a newton metre. A watt is a joule per second. Knowing that lets you rebuild a forgotten formula rather than guess at it.
Dimensional analysis: checking a formula without knowing it
If you are unsure whether kinetic energy is mv squared or mv, check the units. Energy must come out in joules, which are kg m squared per second squared.
This will not give you the whole formula, but it rules out the wrong shape in seconds. If the units of your answer are not the units the question wants, the answer is wrong regardless of how careful the arithmetic was.
Two rules fall out of this and are worth carrying: you can only add quantities with the same units, and units multiply and divide along with the numbers. An equation where one side is metres and the other is seconds is wrong before you check any values.
Significant figures are a claim about precision
Writing 2.5 m says you know the length to the nearest 0.1 m. Writing 2.500 m claims the nearest millimetre. They are different statements even though they are the same number.
Measure a table with a ruler marked in centimetres and report 1.5342 m and you have claimed a precision your instrument cannot deliver. The rule when combining measurements is that the result carries the precision of the least precise input - a chain is as strong as its weakest link, and so is a calculation.
This is also why unit conversion belongs at the start. Convert everything to SI base units before you calculate, not after. Most mixed-unit disasters happen when someone carries centimetres and metres through several steps and loses track of which is which.
Accuracy and precision are different
- Accurate - close to the true value
- Precise - repeatable, tightly clustered
A scale reading 2 kg too heavy every time is precise and inaccurate: consistent, consistently wrong. A scale that scatters randomly around the true value is accurate on average and imprecise.
The distinction matters because the fixes differ. A systematic error - the 2 kg offset - is removed by calibration, and taking more readings will not help at all. Random scatter is reduced by averaging more readings. Knowing which problem you have tells you whether to repeat the experiment or fix the instrument.
Estimating before measuring
Work out roughly what the answer should be before you measure or calculate. If you expect about 3 metres and the calculation says 300, you have made an error by a factor of a hundred - almost certainly a unit conversion - and you have caught it immediately rather than building on it.
This habit prevents more wrong answers than any amount of careful arithmetic, because arithmetic errors are usually small and unit errors are usually enormous.