The key idea
The first chapter of AS, and the one that never stops being examined — because units, vectors and uncertainties appear inside every practical paper and most theory questions until the final A2 exam.
Understand it · 01
Use units to test the physics
An equation is a statement about physical quantities, rather than just a recipe for numbers. A force has units kg m s⁻², so multiplying mass by acceleration gives the correct dimensions. Adding a velocity to an acceleration cannot be meaningful because the quantities have different dimensions. In a calculation, substitute quantities in a consistent set of units before evaluating the expression; a millimetre is 10⁻³ m, whereas a square millimetre is 10⁻⁶ m².
Checking homogeneity can reject an equation, but it cannot establish a missing numerical factor. Both mv² and ½mv² have the dimensions of energy. Similarly, a graph gradient has the units of the vertical axis divided by those of the horizontal axis. Writing those units alongside a gradient helps identify whether it represents a speed, an acceleration, a resistance or a quantity that still needs another factor.
Understand it · 02
Resolve vectors from the diagram
A vector component is the signed projection of a vector onto a chosen axis. First mark the angle on a right-angled triangle. If the angle is measured from the horizontal, the horizontal component is the adjacent side and therefore uses cosine. If the same angle is measured from the vertical, the roles reverse. A negative component means that the vector points against your positive axis; it does not mean that the vector has a negative magnitude.
For several forces, resolve every force along the same two perpendicular directions and add the components separately. Recover the resultant magnitude using Pythagoras. An inverse tangent gives an angle, but the signs of the components determine the quadrant. In an equilibrium problem, both component sums vanish. This is stronger than simply making a sketch look balanced, and is the reason component equations are useful for cables, slopes and interacting charges.
Understand it · 03
Make a measurement claim you can defend
Random variation produces a spread of repeated readings; averaging reduces its effect on an estimate of the mean. A systematic bias, such as a zero offset, can shift every reading together and survives averaging. Measure a suitable range, check the instrument zero, and record the resolution. Do not assume that displaying more decimal places makes an instrument accurate. The size of a sensible uncertainty also depends on the measurement procedure, such as judging the endpoint or aligning a ruler.
For a nonzero value x, percentage uncertainty is 100Δx/|x|. Use the same units for the uncertainty and the reading. Pearson Unit 3 focuses on single-measurement uncertainty; compounding percentage uncertainties belongs to the Unit 6 extension. There, small maximum-uncertainty estimates add absolute uncertainties for sums and differences, and add percentage uncertainties for products and quotients. A power multiplies percentage uncertainty by the magnitude of the exponent. These estimates are not exact statistical confidence intervals.
Key definitions
- Homogeneity
- An equation is homogeneous when every term has the same base units. Homogeneity is necessary for a correct equation, but does not prove it correct.
- Systematic error
- A consistent bias in measurements, such as a zero offset or calibration error. Its size may depend on the measured value; repeating and averaging does not remove it.
- Random error
- Unpredictable variation between repeated measurements under the same conditions. Repeating and averaging can reduce its effect on the estimated mean, but does not remove systematic bias.
- Precision vs accuracy
- Precise readings cluster closely together; accurate readings are close to the true value. A miscalibrated instrument can be precise but not accurate.
Equations, units & conditions
Component resolution
Fx = F cos θ, Fy = F sin θ
Percentage uncertainty
%U = (ΔX ÷ |X|) × 100%
Use a nonzero measured value and a nonnegative absolute uncertainty in the same units.
Powers rule
for Y = Xⁿ: %U(Y) ≈ |n| × %U(X)
A2 extension: school-level maximum-uncertainty approximation for small relative uncertainties.
Worked examples, step by step
Example 1
A force of 20 N acts at 30° above the horizontal. Find its horizontal and vertical components.
Follow the method
- Horizontal (adjacent to the angle): Fx = 20 cos 30° = 17.3 N.
- Vertical (opposite the angle): Fy = 20 sin 30° = 10 N.
Fx ≈ 17.3 N, Fy = 10 N
Practice questions with worked solutions
Try each question on paper before opening the solution. These are original academy questions; the suggested marks are a self-checking guide, not an official exam-board mark scheme. Difficulty labels describe this practice set, not predicted grades.
Build · 3 suggested marks
1. Resolve a cable force
A cable pulls with 50 N at 37° above the horizontal. Calculate its horizontal and vertical components.
Show solution and self-check: question 1
- Fx = 50 cos 37° = 39.93 N.
- Fy = 50 sin 37° = 30.09 N.
Fx = 39.9 N; Fy = 30.1 N.
Self-check · one mark per point
- Use cosine for the horizontal component.
- Use sine for the vertical component.
- Give both components with units.
Build · 2 suggested marks
2. Unit 6 uncertainty extension
A wire radius is (0.40 ± 0.01) mm. Estimate the percentage uncertainty in its cross-sectional area A = πr².
Show solution and self-check: question 2
- Percentage uncertainty in r = (0.01/0.40) × 100 = 2.5%.
- For the square, multiply the percentage uncertainty by 2.
Approximately 5%.
Self-check · one mark per point
- Calculate 2.5% for the radius.
- Apply the powers rule to obtain 5%.
Common mistakes & how to avoid them
- Averaging away a systematic error. Averaging only reduces random error — a zero error survives every repeat. Say which type the error is before saying how to reduce it.
- Forgetting to multiply percentage uncertainty by the power: if T² appears in your equation, the uncertainty in T is doubled on the way through.
- Resolving with the wrong trig function. The component adjacent to the angle takes cos; opposite takes sin. Draw the triangle every time — guessing costs the mark.
- Quoting an answer to more significant figures than the data. Match the least precise value given — usually 2 or 3 s.f.
Understand the why
Questions, explained
Start with the short answer, then follow the reasoning. Each explanation has a permanent link you can share with a classmate or return to when revising.
What is the difference between random and systematic error?
Random error produces unpredictable variation between repeated measurements, while systematic error introduces a consistent bias. Averaging helps estimate the centre of random scatter, but a calibration or zero error remains unless its cause is corrected.
A set of tightly clustered readings can still be inaccurate if the instrument reads too high throughout. Conversely, scattered readings may be centred on the accepted value. Repeating a measurement therefore answers a different question from checking an instrument against a reference. Describe the mechanism when evaluating an experiment: reaction time can vary between trials, while a misaligned zero shifts them together. Neither a larger sample nor more decimal places automatically removes every source of error. Use the spread to assess random variation and an independent reference to investigate bias.
How do you combine uncertainties at A-Level?
For the Unit 6 maximum-uncertainty approximation, add absolute uncertainties in sums or differences and percentage uncertainties in products or quotients. Powers multiply percentage uncertainty by the exponent’s magnitude. These rules assume small relative uncertainties; Unit 3 does not require compounding.
The distinction comes from how the measured quantities affect the result. When two lengths are subtracted, their possible endpoint errors can act in opposite directions, so the maximum absolute uncertainty adds. For a product, it is the fractional changes that approximately add. Treat an exact numerical constant as having no measurement uncertainty. Do not mix these school-level maximum estimates with a statistical root-sum-square calculation unless the question specifically asks for that different model. Always retain units for absolute uncertainty and use the magnitude of a nonzero reading in a percentage.
Unit 6: a calculated density
Mass has 2.0% uncertainty and volume has 3.5% uncertainty. Estimate the percentage uncertainty in density m/V.
- Density is a quotient, so add the percentage uncertainties.
- Estimated maximum percentage uncertainty = 2.0 + 3.5.
Approximately 5.5%.
What does checking homogeneity mean?
Checking homogeneity means expressing every term of an equation in consistent dimensions or SI base units. Terms that are added must match, and both sides must have the same dimensions. A match is necessary but does not prove the equation correct.
Dimensional checks are especially useful after rearranging an equation or deriving an unfamiliar relationship. For example, force multiplied by distance has units kg m² s⁻², the same as energy. A dimensionless factor such as 2 or π cannot be tested this way, so a wrong formula may still be homogeneous. Trigonometric and exponential arguments must also be dimensionless: t/RC is suitable because RC has units of time. Keep dimensional reasoning separate from numerical conversion; centimetres and metres have the same dimensions but cannot be mixed numerically without conversion.
Why can a small extension have a large percentage uncertainty?
Subtracting two similar readings can give a small result with a large percentage uncertainty. The absolute uncertainties add in a maximum estimate, while the denominator is the small difference rather than either original reading.
This issue appears when calculating an extension from two ruler readings or a temperature rise from two thermometer readings. A difference of a few scale divisions can be poorly determined even if both original values look precise. Redesigning the experiment to create a larger measurable change can reduce relative uncertainty, provided the physical model still applies. For a spring, increasing the load only helps while the chosen range remains suitable; crossing into plastic deformation changes the experiment. Do not report an impossible precision just because a calculator shows several digits.
Unit 6: difference of two readings
A ruler gives 12.0 ± 0.1 cm and 12.5 ± 0.1 cm. Estimate the uncertainty in their difference.
- Difference = 12.5 − 12.0 = 0.5 cm.
- Maximum absolute uncertainty = 0.1 + 0.1 = 0.2 cm.
- Percentage uncertainty = 100 × 0.2/0.5.
Difference (0.5 ± 0.2) cm; approximately 40% uncertainty.
How do I choose sine or cosine when resolving a vector?
Use cosine for the component adjacent to the stated angle and sine for the opposite component. The horizontal component is not always the cosine component: that depends on whether the angle is measured from the horizontal or vertical.
Draw the vector as the hypotenuse of a right-angled triangle and label the axes before choosing a function. If a cable’s angle is given from a vertical wall, its vertical component is adjacent to that angle. Add a sign afterwards according to the selected positive direction. For several vectors, resolve all of them onto the same axes; mixing components defined relative to different slopes gives quantities that cannot simply be added. A component cannot have a magnitude greater than the original vector, which provides a useful quick check.
When should I round answers in A-level physics calculations?
Keep extra digits during intermediate calculations, then round the final result to a precision justified by the measurements or the stated uncertainty. Rounding each intermediate quantity too early can shift the final answer appreciably.
A calculation may contain exact counting numbers and measured quantities, so the smallest number of written digits is not always the deciding factor. A factor of two from a formula is exact, while a measured diameter may only have two significant figures. If an uncertainty is supplied, quote the final value to a compatible decimal place rather than adding unsupported digits. Preserve a calculator value when it will be squared or subtracted from a nearby value. Show enough working to make the method clear even when the displayed intermediate result is rounded.
Your learning checklist
Use these goals to check your understanding. Requirements vary by specification, tier and exam year; the official references below define your full course.
- Use SI base units and check equations for homogeneity
- Express values with appropriate significant figures and prefixes (pico to tera)
- Distinguish scalars and vectors; add vectors by drawing and by calculation
- Resolve a vector into perpendicular components
- Distinguish systematic and random errors; estimate single-measurement uncertainty and, for A2 Unit 6, combine uncertainties
Sources & how to cite these notes
Original explanations, illustrations and practice by Dr Desouky Physics Academy. Official specifications guide the course scope; questions and suggested marks on this page are our own revision material.
Reference this page
Dr Desouky Physics Academy. Edexcel IAL Physics: Physical quantities and units. Updated 2026-09-25. https://www.drdesouky.com/revision/a-level-physics/physical-quantities-and-units
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