HookNASA lost a $327 million orbiter to a unit error
On 23 September 1999 the Mars Climate Orbiter fired its engine to slip into orbit behind Mars — and was never heard from again. The failure review found no exotic physics. Lockheed Martin's ground software reported each thruster firing in pound-force seconds; the navigation team's software at JPL read those same numbers as newton seconds. Every trajectory correction for the nine-month cruise was silently wrong by a factor of 4.45. The accumulated drift brought the spacecraft to about 57 km above the surface instead of the planned 226 km; it hit the atmosphere and broke apart. Total mission cost: roughly $327 million. Cause of death: one missing unit conversion that a homogeneity check would have caught in seconds.
AQA puts Measurements and their errors first because it is not really a topic — it is the operating system every other section runs on. Units let you sanity-check any equation you will ever write. Uncertainty is the honest language of all twelve required practicals, and Paper 3 Section A is essentially this section wearing lab gloves. Estimation is the examiner's favourite probe of whether numbers mean anything to you. None of it is hard; all of it is everywhere — which is exactly why it pays marks in every paper, and why NASA's review board would tell you to take it personally.
ModelSix base units build every unit in physics
The SI system rests on base quantities, each with one base unit. A-level physics uses six: mass (kilogram), length (metre), time (second), current (ampere), temperature (kelvin) and amount of substance (mole). Every other unit is derived — assembled from these through the defining equation of the quantity. Force is \(F=ma\), so the newton is \(\text{kg m s}^{-2}\). Energy is force times distance, so the joule is \(\text{kg m}^2\,\text{s}^{-2}\). Power is energy per second: \(\text{kg m}^2\,\text{s}^{-3}\). Strip any equation down to base units and both sides must match — the check is called homogeneity. It cannot prove an equation right (dimensionless factors like the ½ in kinetic energy are invisible to it), but it instantly proves an equation wrong, which makes it worth ten seconds of any exam.
Prefixes then scale units across nature's full range: femto \(10^{-15}\), pico \(10^{-12}\), nano \(10^{-9}\), micro \(10^{-6}\), milli \(10^{-3}\), centi \(10^{-2}\), kilo \(10^{3}\), mega \(10^{6}\), giga \(10^{9}\), tera \(10^{12}\). A nuclear radius is a few femtometres; the Sun sits about 0.15 terametres away. One oddity worth enjoying: the kilogram is the only base unit that arrives with a prefix already attached — the gram is not the base unit of mass. Two habits kill an entire family of errors before they start: convert every prefix to a power of ten before substituting, and carry the units through the algebra rather than bolting them on at the end.
Show that the volt is \(\text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-1}\) in base units. A volt is a joule per coulomb. The joule: \(W=Fs\), so \(\text{J} = (\text{kg m s}^{-2})\times\text{m} = \text{kg m}^2\,\text{s}^{-2}\). The coulomb: \(Q=It\), so \(\text{C}=\text{A s}\). Divide: \(\text{V} = \text{kg m}^2\,\text{s}^{-2} \div \text{A s} = \text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-1}\). Now put it to work. Electric field strength can be written as \(V/d\) or as \(F/Q\). First route: \(\text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-1}\div\text{m}=\text{kg m s}^{-3}\,\text{A}^{-1}\). Second route: \(\text{kg m s}^{-2}\div\text{A s}=\text{kg m s}^{-3}\,\text{A}^{-1}\). Identical — the two definitions are consistent, and you have just performed a homogeneity check exactly the way AQA expects to see one.
ModelRandom scatter, systematic shift — two errors, two different cures
A random error scatters readings unpredictably on both sides of the true value: reaction time on a stopwatch, draughts nudging a top-pan balance, electrical noise in a sensor. You cannot remove it, but you can shrink its effect — repeat and average, time twenty oscillations instead of one, choose an instrument with finer resolution, or plot a graph and let the best-fit line average the scatter for you.
A systematic error shifts every reading the same way: a newton-meter that reads 0.2 N with nothing attached (a zero error), a metre rule with a worn end, a voltmeter out of calibration, a scale read at an angle (parallax). Averaging does nothing here — ten biased readings average to the same bias. The cures are different in kind: check the zero before you start, subtract known offsets, calibrate against a standard, redesign the technique. The give-away in practical questions is a graph that theory says should pass through the origin but cuts an axis instead: random error makes the points wobble about the line; systematic error slides the whole line.
AQA also expects five vocabulary words used precisely. Accuracy: closeness to the true value. Precision: how tightly repeats cluster, regardless of truth. Repeatability: the same experimenter with the same method gets consistent results. Reproducibility: a different experimenter, or different equipment, still gets consistent results. Resolution: the smallest change an instrument can display — a digital balance reading to 0.01 g has a resolution of 0.01 g. Put together: a tight cluster of repeats sitting far from the true value is precise, repeatable and wrong. That combination is the fingerprint of a systematic error, and examiners love asking you to spot it in a results table.
MechanismThe uncertainty algebra AQA actually uses
Every measurement deserves a ±. For a single analogue reading, quote at least half the smallest scale division (a whole division if you judge both ends of a length against a ruler); for a digital display, the instrument's resolution; for repeated readings, half the range of the repeats. Absolute uncertainty wears the same unit as the value — \(l = 0.500 \pm 0.002\) m — while percentage uncertainty divides the absolute uncertainty by the value and multiplies by 100: here 0.4%.
Combining uncertainties follows three rules. Adding or subtracting quantities: add the absolute uncertainties. Multiplying or dividing: add the percentage uncertainties. Raising to a power \(n\): multiply the percentage uncertainty by \(n\) — squaring doubles it, square-rooting halves it. Subtraction is the quiet killer: two large, similar readings subtracted leave a small difference that still carries both absolute uncertainties, so the percentage uncertainty explodes. That is why you measure the diameter of a wire directly with a micrometer rather than as the gap between two nearby marks.
On graphs, uncertainties become error bars, and the uncertainty in a gradient comes from drawing both the best-fit line and the worst acceptable line — the steepest or shallowest line that still passes through every error bar. Percentage uncertainty in the gradient = (difference between the two gradients ÷ best gradient) × 100. Paper 3 asks for some version of this dance almost every year.
A student measures \(g\) with a simple pendulum using \(g = 4\pi^2 l/T^2\). Length \(l = 0.500 \pm 0.002\) m — 0.4%. Twenty complete swings take \(28.4 \pm 0.2\) s, so \(T = 1.42 \pm 0.01\) s — 0.7%. (Timing twenty swings divides the stopwatch uncertainty by twenty; that is the whole point of doing it.) Value: \(g = 4\pi^2 \times 0.500 \div 1.42^2 = 19.74 \div 2.016 = 9.79\ \text{m s}^{-2}\). Uncertainty: \(T\) is squared, so its percentage counts double — total = 0.4% + 2(0.7%) = 1.8%. Absolute: 1.8% of 9.79 = 0.18. Result: \(g = 9.8 \pm 0.2\ \text{m s}^{-2}\) — uncertainty rounded to one significant figure, value rounded to match its decimal places, and the accepted 9.81 sits comfortably inside the interval. That final formatting sentence is routinely a mark on its own.
DataPowers of ten: the physicist's sanity check
An order of magnitude is a value rounded to the nearest power of ten, and its point is comparative. An atom is about \(10^{-10}\) m across; its nucleus is about \(10^{-14}\) m — four orders of magnitude smaller, a fly in a cathedral. Examiners use order-of-magnitude questions to find out whether numbers carry meaning for you, so bank a handful of anchors: atomic diameter \(10^{-10}\) m, nuclear diameter \(10^{-14}\) m, wavelength of green light \(5\times10^{-7}\) m, density of water \(10^{3}\ \text{kg m}^{-3}\), atmospheric pressure \(10^{5}\) Pa, a person's mass \(10^{2}\) kg.
Estimation turns the same instinct into marks: break the target into factors you can guess to one significant figure, multiply, and state your assumptions. AQA has asked for the kinetic energy of a sprinter, the pressure under a stiletto heel, the power of a kettle — quantities nobody memorises, all reachable in three lines. Your answer is judged on method and plausibility, not on matching a hidden 'true' value. Working to one significant figure is not laziness; it is the correct register. An estimate quoted to three significant figures is lying about what it knows — exactly the crime the Mars software committed with false confidence in the wrong units.
Estimate the weight of the air in your bedroom. Assume a room 4 m × 3 m × 2.5 m, volume \(30\ \text{m}^3\). Air density near sea level is about \(1.2\ \text{kg m}^{-3}\), so the mass is roughly \(30 \times 1.2 = 36\) kg. Weight \(= mg \approx 36 \times 9.8 \approx 350\) N — call it \(4\times10^{2}\) N, order of magnitude \(10^{2}\) N. State the assumptions (room dimensions, uniform density) and stop at one or two significant figures. The surprise — the air in your room outweighs a large dog — is precisely the sense of scale the question is probing.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Three formatting rules each carry marks. Give your answer to the same number of significant figures as the least precise piece of data — an answer to five figures from three-figure data is wrong even when the digits are right. Quote uncertainties to one significant figure (two at most) and round the value to match its decimal places: \(9.8 \pm 0.2\), never \(9.789 \pm 0.2\). And put a unit on every numerical answer; in 'show that' questions, work to one more significant figure than the target and state the comparison explicitly.
Paper 3 Section A is where 3.1 concentrates: reading error bars, drawing best and worst acceptable lines, and turning the pair into a percentage uncertainty in the gradient. When asked how to reduce uncertainty, name the quantity, the instrument and the technique — 'measure twenty oscillations and divide by twenty', 'use a micrometer with 0.01 mm resolution on the wire diameter'. Generic answers about being careful or avoiding human error score zero, and AQA's examiner reports say so almost annually.
The AO structure rewards the same discipline. Around a third of marks are AO1 recall, over 40% are AO2 application, and roughly a quarter are AO3 analysis — and at least 40% of all physics marks involve Level 2 maths. The classic AO3 move in this section: compare your result with the accepted value through the uncertainty. 'The interval 9.8 ± 0.2 m s⁻² includes 9.81, so the result is consistent with the accepted value' is the exact sentence the mark scheme is waiting for; a bare percentage difference without the comparison is half an answer.