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Estimating Physical Quantities

Key idea: How to make quick, reasonable order-of-magnitude estimates using SI units, standard form, and simple physical models (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Estimate physical quantities and check the reasonableness of results.

1. Definitions (Must Know)

  • Estimate: an approximate value based on reasonable assumptions and simplified models.
  • Order of magnitude: the power-of-ten scale of a value. Write the estimate in standard form first, then state the power of ten; if a question explicitly asks for the nearest power of ten, use the geometric boundary square root of 10 ≈ 3.16.
  • Standard form: a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
  • Bound: an upper or lower limit for a quantity when you are not sure of the exact value.

2. Key Ideas (What Earns Marks)

  • Convert to SI units first (m, kg, s, A, K).
  • Round typical values to 1 s.f. while estimating to keep arithmetic fast.
  • State your assumptions (typical values, approximations, shape/model).
  • Use a single, clear model equation (e.g. p = F/A, t = L/c, m = ρ V).
  • Quote a final answer with a sensible unit and an order of magnitude (not lots of s.f.).

3. Detailed Explanations

A. A good estimation workflow

  1. Pick a simple model (what equation connects the quantities?).
  2. Choose typical values (use round numbers).
  3. Calculate in standard form.
  4. State the order of magnitude and sanity-check it.

Mini-example (pressure): If p = F/A and you estimate F∼ 10³ N and A∼ 10⁻² m², then p ∼ 10³/10⁻² = 10⁵ Pa

B. Bounding (when you’re unsure)

If a value is uncertain, choose a lower and upper bound, estimate both, and show the range your answer could lie in.

4. Common Mistakes

  • Forgetting unit conversions (e.g. using cm instead of m).
  • Giving too many significant figures (an estimate should not look “exact”).
  • Choosing values that are not physically reasonable (e.g. a human mass of 5 kg).
  • Confusing “order of magnitude” with “1 s.f.” (one is a power-of-ten scale; the other is a rounded numerical value).

5. Exam Tips

  • Use the symbol ≈ to show you are estimating.
  • Write units on every step to catch mistakes early.
  • A one-line reasonableness check is often enough (e.g. “this is about atmospheric pressure”).
  • Use data-booklet-friendly constants when appropriate (e.g. g ≈ 10 m s⁻², c ≈ 3 × 10⁸ m s⁻¹).
  • Related: Dimensional Analysis (units check) and Uncertainty & Error Propagation (how accurate your estimate needs to be).

6. Worked Examples

Modelled example 1

Estimate the pressure a person exerts standing on one foot

Core

Problem

Estimate the pressure exerted by a person standing on one foot, stating reasonable assumptions.
Study the worked solution

  1. Choose a model

    Method

    Use p = F/A with F ≈ mg.

    Reason

    Standing pressure is weight divided by contact area.

    Working

    Assume m ≈ 60 kg and g ≈ 10 m s⁻².

  2. Estimate contact area

    Method

    Use 200 cm² = 2 × 10⁻² m².

    Reason

    A stated plausible foot area makes the estimate auditable.

    Working

    A ≈ 2 × 10⁻² m².

  3. Calculate and classify

    Method

    Obtain 3 × 10⁴ Pa, order 10⁴ Pa.

    Reason

    F ≈ 600 N.

    Working

    p ≈ 600/(2 × 10⁻²) = 3 × 10⁴ Pa

Guided practice 2

Estimate the time for light to cross a 5 m room

About 4 min

Problem

Estimate the time for light to cross a 5 m room.

Try this before viewing the solution

Unit: s

Hints

Hint 1: use a data-booklet constant
Take c ≈ 3 × 10⁸ m s⁻¹.
View solution step by step

  1. Choose the relation

    Method

    Use t = L/c.

    Reason

    Light travels at approximately constant speed across the room.

    Working

    t ≈ 5/(3 × 10⁸)

  2. Evaluate

    Method

    Obtain 1.7 × 10⁻⁸ s, order 10⁻⁸ s.

    Reason

    The coefficient 1.7 is below the nearest-order boundary square root of 10.

    Working

    t ≈ 1.7 × 10⁻⁸ s.

Common misconception 3

Estimate the gravitational potential energy gained by lifting a 2 kg object by 1.5 m

Find and correct the mistake

Learner claim

Lifting 2 kg by 1.5 m gives about 30 J. A learner calls this order 10² J because 30 is closer arithmetically to 100 than 10. Diagnose the classification.

Try this before viewing the solution

Nearest power of ten

View solution step by step

  1. Estimate the energy

    Method

    Use Δ Eₚ = mgh with g ≈ 10.

    Reason

    The simple near-Earth model is sufficient.

    Working

    Δ Eₚ ≈ 2(10)(1.5) = 30 J = 3.0 × 10¹ J

  2. Apply the logarithmic boundary

    Method

    Choose 10¹ J.

    Reason

    3.0 < square root of 10 ≈ 3.16, so 3.0 × 10¹ is nearest to 10¹ on a log scale.

    Working

    Order of magnitude: 10¹ J.

Examiner practice 4

Estimate the average power when climbing stairs

4 marks

Examination question

A 70 kg student climbs 4 m vertically in 6 s. Estimate average power and its order of magnitude. [4 marks]

Try this before viewing the solution

View solution step by step

  1. Choose the energy model

    1 mark

    Method

    Use Δ E ≈ mgh.

    Reason

    Vertical climbing increases gravitational potential energy.

    Working

    Δ E ≈ mgh.

  2. State an estimate

    1 mark

    Method

    Take g ≈ 10 m s⁻².

    Reason

    A one-significant-figure constant suits the estimate.

    Working

    g ≈ 10 m s⁻².

  3. Calculate power

    1 mark

    Method

    Obtain 4.7 × 10² W.

    Reason

    P = Δ E/t.

    Working

    P ≈ 70(10)(4)/6 = 4.7 × 10² W

  4. State the order

    1 mark

    Method

    Give 10² W.

    Reason

    The estimate is hundreds of watts.

    Working

    Order: 10² W.

Challenge 5

Estimate the thickness of one sheet of paper (stack method)

Minimal support

Independent transfer

A stack of 200 sheets is 18 mm thick. Estimate one-sheet thickness in metres, give its nearest order of magnitude, and explain why stacking helps.

Try this before viewing the solution

Hints

Hint 1: average before classifying
Divide the stack thickness by sheet count, convert to metres, then write standard form.
View solution step by step

  1. Find average sheet thickness

    Method

    Divide by 200.

    Reason

    The stack amplifies a thickness too small to measure reliably alone.

    Working

    t ≈ 18/200 = 0.09 mm

  2. Convert and classify

    Method

    Obtain 9 × 10⁻⁵ m, nearest order 10⁻⁴ m.

    Reason

    0.09 mm = 9 × 10⁻⁵ m and coefficient 9 is above square root of 10.

    Working

    t ≈ 9 × 10⁻⁵ m∼10⁻⁴ m.

7. Mind Stretchers

Mind stretcher 1: Estimate the mass of air in a classroomExtension

Show Answer

Assume a classroom is 8 m × 6 m × 3 m so volume V ≈ 8 × 6 × 3 ≈ 1.4 × 10² m³

Using air density ρ ≈ 1.2 kg m⁻³: m = ρ V ≈ 1.2 × 1.4 × 10² ≈ 1.7 × 10² kg

Order of magnitude: 10² kg.

Mind stretcher 2: Estimate the gravitational force between two studentsExtension

Two students each have mass ≈ 60 kg and stand about 1 m apart. Estimate the gravitational force between them.

Show Answer

Use Newton’s law: F ≈ Gm²/r²

Estimate with G ≈ 6.7 × 10⁻¹¹ N m²kg⁻², m ≈ 60 kg, r ≈ 1 m: F ≈ 6.7 × 10⁻¹¹ × 60² ≈ 6.7 × 10⁻¹¹ × 3.6 × 10³ ≈ 2.4 × 10⁻⁷ N

Order of magnitude: 10⁻⁷ N.

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027