Quantities & Measurement: units, uncertainty and vectors
Key idea: Connect unit reasoning, realistic estimates, measurement quality and vector components into one reliable problem-solving routine.
Before you start: Dimensional AnalysisUncertainty
By the end, you can
- Use SI base quantities, prefixes and derived units, and test equations for dimensional homogeneity.
- Make defensible order-of-magnitude estimates.
- Distinguish random and systematic errors and propagate stated uncertainties without rigorous statistics.
- Distinguish scalars from vectors and resolve, add and subtract coplanar vectors.
Starting-point self-check
1. Check your starting point
Attempt all five prompts without notes. Mark the first reasoning step you could not justify. Use the recorded topic diagnostic above when you want scoring and a personalised repair plan.
Question 1
Convert 3.5 μm to metres. Then name the SI base units for mass, time, current and temperature.
Check the model response
3.5 μm = 3.5 × 10⁻⁶ m. The requested base units are kilogram (kg), second (s), ampere (A) and kelvin (K).
Question 2
Express the joule in SI base units and decide whether s = ut + ½at is dimensionally homogeneous.
Check the model response
1 J = 1 kg m² s⁻². The equation is not homogeneous: ut has dimension L, but at has dimension LT⁻¹. The uniformly accelerated term requires at².
Question 3
Estimate the mass of air in a classroom measuring about 8 m by 6 m by 3 m. Use air density 1.2 kg m⁻³ and state an appropriate order of magnitude.
Check the model response
Volume ≈ 144 m³, so mass ≈ 1.2 × 144 = 173 kg. A defensible order of magnitude is 10² kg.
Question 4
Repeated readings scatter around a value, but a balance also reads +0.04 kg when empty. Classify both effects and state which limits precision and which limits accuracy.
Check the model response
The scatter is random error and limits precision. The positive zero error is systematic and shifts every reading, limiting accuracy unless corrected.
Question 5
A 12 N vector is directed 35° above the positive x-axis. Resolve it into perpendicular components.
Check the model response
Fx = 12 cos 35° = 9.8 N and Fy = 12 sin 35° = 6.9 N, using positive x and positive y as the declared directions.
repair
2. Repair the four common breaks
Use the repair matching each diagnostic error, then rework that prompt.
Check this idea
Misconception: A prefix changes only the written symbol, not the numerical scale.
Repair: Replace every prefix by its power of ten before calculating. Case matters: m is 10⁻³ while M is 10⁶.
Check this idea
Misconception: Matching dimensions proves a proposed physical equation is correct.
Repair: Homogeneity is necessary, not sufficient. It can reject an equation with mismatched dimensions but cannot prove numerical factors or the physical model.
Check this idea
Misconception: A physical estimate is a guess that should copy an exact reference value.
Repair: A defensible estimate states reasonable dimensions, rates or material properties, calculates from them and checks the resulting order of magnitude.
Check this idea
Misconception: More decimal places remove uncertainty and improve accuracy.
Repair: Displayed digits do not remove random scatter or systematic offset. Quote precision justified by the instrument and propagate the stated uncertainty.
Check this idea
Misconception: Vector magnitudes may be added or subtracted without directions.
Repair: Choose axes, resolve each vector with signs, combine corresponding components, then recover magnitude and direction.
worked example
3. Follow four worked models
Track units, assumptions, uncertainty rules and directions explicitly.
Model 1
Show that power has SI base units kg m² s⁻³, starting from P = E/t.
Check the model response
Energy has unit J = kg m² s⁻². Dividing by time gives kg m² s⁻²/s = kg m² s⁻³.
Model 2
Estimate the energy transferred by a 2.0 kW kettle operating for 3 minutes.
Check the model response
Use E = Pt with 2.0 kW = 2.0 × 10³ W and 3 min = 180 s. E = 3.6 × 10⁵ J, sensibly of order 10⁵ J.
Model 3
A rectangle has L = (2.40 ± 0.02) m and W = (1.20 ± 0.01) m. Find its area and absolute uncertainty.
Check the model response
A = 2.88 m². For multiplication, add relative uncertainties: 0.02/2.40 + 0.01/1.20 = 0.0167. Thus ΔA ≈ 0.0167 × 2.88 = 0.048 m², so A = (2.88 ± 0.05) m².
Model 4
Add 8 N east to 6 N north, then subtract the 6 N north vector from the 8 N east vector.
Check the model response
With east as +x and north as +y, the sum is (8, 6) N: magnitude 10 N at 36.9° north of east. The difference is (8, −6) N: magnitude 10 N at 36.9° south of east.
guided practice
4. Guided practice
Use each hint for the setup only; finish the reasoning before opening the response.
Question 1
Convert 7.2 GW to watts and write the pascal in SI base units.
Hint: Replace giga by 10⁹; use pressure = force/area.
Check the model response
7.2 GW = 7.2 × 10⁹ W. Pa = N m⁻² = kg m s⁻² m⁻² = kg m⁻¹ s⁻².
Question 2
Check the homogeneity of p = ρgh using SI base dimensions.
Hint: Write density as mass per volume.
Check the model response
[ρgh] = (M L⁻³)(L T⁻²)(L) = M L⁻¹ T⁻², the same dimensions as pressure.
Question 3
For Q = (a − b)c, state the uncertainty procedure when a, b and c each have absolute uncertainties.
Hint: Use the addition rule before the multiplication rule.
Check the model response
First find d = a − b and add absolute uncertainties: Δd = Δa + Δb. Then Q = dc and add relative uncertainties: ΔQ/Q = Δd/|d| + Δc/|c|.
Question 4
Resolve a 20 m displacement at 30° west of north into east and north components.
Hint: Declare east +x and north +y before assigning signs.
Check the model response
East component = −20 sin 30° = −10 m. North component = 20 cos 30° = 17.3 m.
independent practice
5. Independent practice
Write complete solutions without using the repair notes.
Question 1
Convert 450 nm to metres and 0.75 MJ to joules. Derive the SI base unit of electric charge from Q = It.
Check the model response
450 nm = 4.50 × 10⁻⁷ m; 0.75 MJ = 7.5 × 10⁵ J; charge has base unit A s.
Question 2
A proposed pendulum relation is T = 2π√(l/g). Test its dimensional homogeneity and explain what that test cannot establish.
Check the model response
[l/g] = L/(L T⁻²) = T², so its square root has dimension T. The relation is homogeneous, but this does not prove the factor 2π or that the model applies.
Question 3
A speed is calculated from d = (5.00 ± 0.05) m and t = (2.00 ± 0.02) s. Find the speed and its percentage uncertainty.
Check the model response
v = 2.50 m s⁻¹. Percentage uncertainty = (0.05/5.00 + 0.02/2.00) × 100% = 2.0%, so v = (2.50 ± 0.05) m s⁻¹.
Question 4
Two coplanar forces are A = (−3, 4) N and B = (5, −2) N. Find A + B and A − B, giving component and magnitude forms.
Check the model response
A + B = (2, 2) N with magnitude 2.83 N. A − B = (−8, 6) N with magnitude 10.0 N.
Practice exit check
6. Practice assessment
Use this as extra closed-book practice, then complete the separate recorded assessment in your plan.
Question 1
Write 6.2 GJ in joules and in SI base units. Then derive the base unit of power.
Check the model response
6.2 GJ = 6.2 × 10⁹ J = 6.2 × 10⁹ kg m² s⁻². Power has base unit kg m² s⁻³.
Question 2
Estimate the mass of water in a domestic bathtub, stating dimensions or volume assumptions and an order of magnitude.
Check the model response
For example, 1.5 m × 0.6 m × 0.25 m ≈ 0.225 m³. With water density 1000 kg m⁻³, mass ≈ 225 kg, of order 10² kg. Other defensible assumptions earn the same conclusion.
Question 3
Times are 1.42 s, 1.47 s and 1.44 s, while the timer has a +0.05 s zero error. Estimate the corrected mean with a random uncertainty from half the range.
Check the model response
Measured mean = 1.443 s; subtract the +0.05 s zero error to get 1.393 s. Half-range = (1.47 − 1.42)/2 = 0.025 s, quoted as about 0.03 s. Corrected result: (1.39 ± 0.03) s.
Question 4
Let A be 8 N east and B be 6 N north. Determine A + B and A − B, including directions.
Check the model response
A + B has magnitude 10 N at 36.9° north of east. A − B has magnitude 10 N at 36.9° south of east.
Re-test practice
7. Delayed re-test practice
Return after at least three days. These contexts and numbers differ from the assessment. The recorded plan enforces the delay and uses a separate re-test family for selected-response skill-group evidence.
Question 1
Derive the SI base unit of pressure and use dimensions to test p = ρgh.
Check the model response
Pressure has unit kg m⁻¹ s⁻². [ρgh] = (kg m⁻³)(m s⁻²)(m) = kg m⁻¹ s⁻², so the equation is homogeneous.
Question 2
Estimate the number of heartbeats in one day for a resting rate of about 70 min⁻¹ and give the order of magnitude.
Check the model response
70 × 60 × 24 = 100800 ≈ 1.0 × 10⁵ beats, so the order of magnitude is 10⁵.
Question 3
A cylinder has diameter (2.50 ± 0.02) cm and length (10.0 ± 0.1) cm. Calculate its volume and percentage uncertainty using V = πd²l/4.
Check the model response
V = 49.1 cm³. Relative uncertainty = 2(0.02/2.50) + 0.1/10.0 = 0.026, or 2.6%. Thus V ≈ (49.1 ± 1.3) cm³.
Question 4
A displacement of 15 m at 120° anticlockwise from east is followed by 4 m east. Find the resultant components, magnitude and direction.
Check the model response
The first displacement is (15 cos 120°, 15 sin 120°) = (−7.5, 13.0) m. Adding (4, 0) gives (−3.5, 13.0) m, magnitude 13.5 m at about 105° anticlockwise from east.