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AS · Practice questions · Quantities and units

Reduce it to the base.

Six original Cambridge-style questions: deriving units from base units, using homogeneity to pin down an unknown power, converting prefixes, and judging what a units check can and cannot prove.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
Keep these straight

Magnitude and unit, every time.

01
Analysis
[2 marks]

State what is meant by a physical quantity, and use the quantity "a current of 0.50 A" to identify its two parts.

  • A physical quantity is a magnitude (a numerical value) together with a unit. ✓
  • For 0.50 A the magnitude is 0.50 and the unit is the ampere (A). ✓
02
Analysis
[3 marks]

The joule is the SI unit of energy. Starting from a suitable defining equation, express the joule in terms of SI base units.

  • Use energy = force × distance, so 1 J = 1 N × 1 m. ✓
  • The newton in base units is kg m s⁻². ✓
  • So 1 J = kg m s⁻² × m = kg m² s⁻². ✓
03
Analysis
[3 marks]

The drag force on a sphere falling through a liquid is modelled by F = k r v, where r is a radius, v is a speed and k is a constant. Determine the SI base units of k.

  • Rearrange: k = F / (r v). ✓
  • Units: F is kg m s⁻², r is m, v is m s⁻¹. So k = (kg m s⁻²) / (m × m s⁻¹). ✓
  • This simplifies to kg m⁻¹ s⁻¹. ✓
04
Analysis
[2 marks]

A wavelength is recorded as 0.0000006 m. Express this in nanometres and state the prefix used.

  • 0.0000006 m = 6 × 10⁻⁷ m = 600 × 10⁻⁹ m. ✓
  • Since nano means 10⁻⁹, this is 600 nm. ✓
05
Analysis
[2 marks]

Estimate, to the nearest order of magnitude, the number of seconds in a human lifetime. Show your reasoning and give the answer with its unit.

  • One year is about 3 × 10⁷ s (365 × 24 × 3600 ≈ 3.15 × 10⁷). ✓
  • A lifetime of order 80 years gives about 80 × 3 × 10⁷ ≈ 2.5 × 10⁹, so of order 10⁹ s. ✓
06
Analysis
[3 marks]

A student checks the equation v² = u² + 2as by units and finds it homogeneous. They conclude the equation must be correct. Explain why their reasoning is flawed, and state what the check does establish.

  • Homogeneity is necessary but not sufficient: a units check cannot detect a wrong dimensionless factor (for example a missing or extra pure number), so a homogeneous equation can still be wrong. ✓
  • What it does establish: if the units had not matched, the equation would certainly be wrong. A successful check only shows the equation is not ruled out by units. ✓
  • (Here each term reduces to m² s⁻², so it passes the test, which is consistent with the equation being correct but does not prove it.) ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.