Free Pearson Edexcel International A Level Physics lesson

Measurement, units and evidence

Give every number a meaning and every conclusion an uncertainty.

What you’ll learn

  • Convert and check units, including squared and cubed prefixes.
  • Distinguish random scatter from systematic effects and combine simple uncertainties.
  • Choose useful tables, graphs and controls to test a physical model.

A physical quantity is more than a number

A measured value combines a number with a unit. Reporting a trolley speed as 2.4 leaves the result incomplete: 2.4 m s⁻¹ means a different motion from 2.4 km h⁻¹. Write raw measurements with their original units, then convert consistently before substitution. The SI base units used most in this unit are metre, kilogram and second. Derived units reveal relationships: one newton is one kg m s⁻², and one joule is one N m.

A prefix multiplies the whole quantity it belongs to. Recall these seven prefixes with their case-sensitive symbols: giga (G), 10⁹; mega (M), 10⁶; kilo (k), 10³; centi (c), 10⁻²; milli (m), 10⁻³; micro (μ), 10⁻⁶; and nano (n), 10⁻⁹. For example, 2.0 GPa = 2.0 × 10⁹ Pa = 2000 MPa, while 400 μm = 0.400 mm and 600 nm = 6.00 × 10⁻⁷ m. Powers apply to the prefix too: 1 cm² = 10⁻⁴ m² and 1 cm³ = 10⁻⁶ m³. Write the conversion as a ratio equal to one if you are unsure. Standard form keeps a number between 1 and 10 multiplied by a power of ten, so 0.00045 m becomes 4.5 × 10⁻⁴ m.

Check a proposed equation before calculating. The two sides must have the same dimensions, and terms added together must share units. Dimensional agreement is necessary but cannot determine a factor such as ½ or prove a model valid. Estimate the order of magnitude, keep extra digits during working, then round the final result to precision supported by the input measurements. Exact counts and defined conversion factors do not limit significant figures.

1 N = 1 kg m s⁻² 1 J = 1 N m 1 cm³ = (10⁻² m)³ = 10⁻⁶ m³

Symbols and units

  • N, J, kg, m and s denote newton, joule, kilogram, metre and second.
  • The prefix centi means 10⁻²; a cubed length applies the cube to its conversion factor.

How the equation explains the idea

  1. Force has dimensions mass × acceleration, giving 1 N = 1 kg m s⁻²; work is force × displacement, giving 1 J = 1 N m.
  2. Since 1 cm = 10⁻² m, multiply this relation three times: 1 cm³ = 10⁻⁶ m³. A volume conversion cannot use the length factor alone.

Worked example

One conversion in each dimension

  1. A rectangular insert measures 20.0 mm × 10.0 mm × 5.00 mm.
  2. Convert each length: 0.0200 m × 0.0100 m × 0.00500 m.
  3. The volume is 1.00 × 10⁻⁶ m³, equivalent to 1.00 cm³ or 1000 mm³.

The volume conversion factor is cubed; it is not the length conversion factor.

Keep this idea. Carry units through the working; convert powers of a length with the same power of its prefix.

Separate scatter, bias and uncertainty

Repeated measurements rarely agree exactly. Random effects cause readings to scatter, whereas a systematic effect shifts results in a consistent way, such as an uncorrected zero on a balance. Precision describes how closely readings cluster; accuracy concerns closeness to the true value. A very repeatable result can be inaccurate. Repeating and averaging can reduce random influence on the mean, but will not remove a zero error shared by every reading.

State how uncertainty was estimated. Instrument resolution is only one contribution; reaction time, alignment or the definition of the measured endpoints may dominate. A half-range estimate from repeats is a useful simple description of spread, not a universal confidence interval. Quote an absolute uncertainty in the measured unit, or percentage uncertainty 100 × absolute uncertainty/value. For a difference or sum, a conservative estimate adds absolute uncertainties. For products and quotients, add percentage uncertainties; a power multiplies percentage uncertainty by its absolute power.

These simple propagation rules give a cautious approximate bound for small uncertainties. They are not a claim that every effect is independent or normally distributed. Check for correlated effects: a fixed ruler zero may cancel when subtracting two readings from the same scale, while uncertain endpoint judgement still affects the difference. Do not erase an unusual reading merely because it is inconvenient; repeat it, investigate a cause and record any justified exclusion. Report the result and uncertainty to compatible decimal places.

Percentage uncertainty = 100 × δx/|x| For z = x²: percentage uncertainty in z ≈ 2 × that in x

Symbols and units

  • x is a measured quantity and δx is its positive absolute uncertainty in the same unit.
  • δx/|x| is dimensionless fractional uncertainty; multiply by 100 for percentage uncertainty.

How the equation explains the idea

  1. A fixed δx is a larger fraction when |x| is small: 0.1 mm uncertainty in 2.0 mm is 5%, but in 10.0 mm it is 1%.
  2. For z = x², a small change gives δz ≈ 2|x|δx. Dividing by z = x² gives δz/|z| ≈ 2δx/|x|; this is a small-uncertainty approximation.

Worked example

Two ruler readings define one extension

  1. The initial reading is 12.0 ± 0.1 cm and the final reading 16.4 ± 0.1 cm; treat their reading uncertainties conservatively.
  2. Extension = 16.4 − 12.0 = 4.4 cm. Absolute uncertainty = 0.1 + 0.1 = 0.2 cm.
  3. Percentage uncertainty = 100 × 0.2/4.4 ≈ 4.5%. A shared fixed zero offset would cancel separately.

Report extension as 4.4 ± 0.2 cm under the stated uncertainty model.

Keep this idea. Identify the source of uncertainty before choosing repeats, calibration or a different method.

Turn measurements into a test of a model

Choose a variable to change, a response to measure and quantities to keep constant because they could affect that response. Record raw readings in a table with units in the headings and consistent precision. Repeat over a useful range, including values that can distinguish competing models. For example, doubling a pull and doubling the mass at the same time cannot isolate the effect of force on acceleration.

A graph uses all the data more clearly than calculating a ratio from one point. Choose labelled axes and a scale that uses the available space. Plot points accurately; draw a suitable best-fit line or curve rather than joining every fluctuation. Use a large triangle on a straight best-fit line for its gradient, and read an intercept with its physical meaning. Proportionality requires a straight line through the appropriate zero; a nonzero intercept may represent an offset, a physical initial value or evidence that a simpler model is incomplete.

Use error bars or a stated spread estimate to judge agreement. Compare a model with the pattern and size of residuals, the measured-minus-predicted differences. Random residuals around zero are consistent with the model over the tested range; a curved pattern suggests a missing effect. Agreement within uncertainty supports a model but never proves it exact. When estimating a gradient range, draw the steepest and shallowest plausible straight lines consistent with the uncertainty regions, then state the method and bounds.

y = mx + c Gradient m = Δy/Δx Residual = measured y − model y

Symbols and units

  • x and y denote the plotted quantities; their units come from the axis labels.
  • m is gradient, with units y-unit per x-unit; c is the y-intercept in y-units. Here m is not mass.

How the equation explains the idea

  1. Subtract y₁ = mx₁ + c from y₂ = mx₂ + c to cancel the intercept: Δy = mΔx, hence m = Δy/Δx.
  2. A residual is measured y minus predicted mx + c. A curved pattern of residuals suggests the straight-line model misses a systematic relationship, rather than proving a particular cause.
A straight line is not necessarily direct proportion

Conditions. Illustrative measurement models x = 2F and xindicated = 2F + 2, with force in N and extension in mm. The second model adds a constant 2 mm zero offset, with no random scatter.

Applied force / N; Indicated extension / mm.

Both slopes are 2 mm N⁻¹, equivalent to a stiffness of 500 N m⁻¹. Only the zero-intercept line is direct proportion. Correcting a constant zero offset moves a line vertically without changing its gradient.

Open the interactive lesson to view the plotted curves and their coordinate tables.

Worked example

A line with a physical intercept

  1. A best-fit velocity–time line passes through (0.50 s, 2.0 m s⁻¹) and (2.50 s, 6.0 m s⁻¹).
  2. Gradient = (6.0 − 2.0)/(2.50 − 0.50) = 2.0 m s⁻².
  3. Using v = at + u gives u = 2.0 − 2.0 × 0.50 = 1.0 m s⁻¹.

The intercept represents initial velocity; dividing v by t would not give the correct acceleration.

Keep this idea. Interpret gradient, intercept and uncertainty together; a graph should test an assumption, not just decorate a table.

Open the interactive lesson for free practice and reviewed three-question samples when available. New full question sets and timed unit checks are in preparation. Existing complete Physics revision packs retain paid access; existing free teaching PDFs remain free.

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