SI Units, Prefixes and Estimation — AQA A-level Physics

SPEC 3.1.1, 3.1.3 - MEASUREMENTS AND THEIR ERRORS

SI Units, Prefixes & Estimation

Master the foundation of all physics calculations: base units, prefix conversions, and order-of-magnitude reasoning

Why SI Units?

The Système International d'Unités (SI) provides a universal language for measurement. Without standardised units, scientific communication would be impossible.

Definition

SI Base Units are the fundamental units from which all other units are derived. They cannot be expressed in terms of other SI units.

Key Point

Every physical quantity in A Level Physics can be expressed using combinations of the six base units you need to know.

The SI Base Units

QuantityUnit NameSymbol

Lengthmetrem

Masskilogramkg

Timeseconds

Electric CurrentampereA

TemperaturekelvinK

Amount of Substancemolemol

Warning

The kilogram is the only base unit with a prefix. Don't confuse it with the gram—the base unit is kg, not g.

SI Prefixes

Prefixes allow us to express very large or very small quantities without writing out all the zeros.

PrefixSymbolPower

teraT10¹²

gigaG10⁹

megaM10⁶

kilok10³

centic10⁻²

PrefixSymbolPower

millim10⁻³

microμ10⁻⁶

nanon10⁻⁹

picop10⁻¹²

femtof10⁻¹⁵

Standard Form

Standard form is used to represent very large or very small numbers concisely.

Large Numbers

Move decimal left → positive power

6,370,000 = 6.37 × 10⁶

Small Numbers

Move decimal right → negative power

0.0004 = 4 × 10⁻⁴

Worked Example

Standard Form Conversions01

(a) Write 1,000,000 in standard form.

(b) Write 0.0004 in standard form.

(c) Write 6,370,000 m in standard form.

(a)

1,000,000 → move decimal 6 places left → 1 × 10⁶

(b)

0.0004 → move decimal 4 places right → 4 × 10⁻⁴

(c)

6,370,000 → move decimal 6 places left → 6.37 × 10⁶ m

Converting Prefixes

Method

To convert between prefixes, multiply by 10 raised to the difference in powers.

new value = old value \times 1 0^{(old power - new power)}

Step-by-Step

Identify the power of 10 for each prefix

Calculate: (starting power) − (target power)

Multiply the number by 10 raised to this difference

Quick Check

Going to a smaller prefix → number gets bigger

Going to a larger prefix → number gets smaller

Worked Example

Prefix Conversions02

(a) Convert 4.7 μm to nm.

(b) Convert 0.35 MW to kW.

(c) Convert 250 ps to ns.

(a)

μ = 10⁻⁶, n = 10⁻⁹ → difference: (−6) − (−9) = +3

•

4.7 × 10³ = 4700 nm

(b)

M = 10⁶, k = 10³ → difference: 6 − 3 = +3

•

0.35 × 10³ = 350 kW

(c)

p = 10⁻¹², n = 10⁻⁹ → difference: (−12) − (−9) = −3

•

250 × 10⁻³ = 0.25 ns

Converting Units

The specification requires you to convert between different unit systems.

Key Conversions

ConversionRelationshipHours to seconds1 hour = 3600 sElectron volts to joules1 eV = 1.6 × 10⁻¹⁹ JKilowatt hours to joules1 kW h = 3.6 × 10⁶ J = 3.6 MJ

Common Error

When converting "2 hours 20 minutes" to seconds:

✗ Don't use 2.2 hours (this gives 7920 s — wrong!)

✓ Convert separately: (2 × 3600) + (20 × 60) = 7200 + 1200 = 8400 s

Worked Example

Unit Conversions03

(a) Convert 76 MeV to joules.

(b) Convert 1 kW h to joules.

(a)

First convert MeV to eV: 76 MeV = 76 × 10⁶ eV

•

Then convert to joules: 76 × 10⁶ × 1.6 × 10⁻¹⁹ J

•

= 1.22 × 10⁻¹¹ J

(b)

1 kW = 1000 J s⁻¹, 1 hour = 3600 s

•

1 kW h = 1000 × 3600 = 3.6 × 10⁶ J = 3.6 MJ

Derived Units

All other units are built from combinations of the base units using defining equations.

QuantityUnitSymbolIn Base Units

ForcenewtonNkg m s⁻²

EnergyjouleJkg m² s⁻²

PowerwattWkg m² s⁻³

PressurepascalPakg m⁻¹ s⁻²

ChargecoulombCA s

VoltagevoltVkg m² s⁻³ A⁻¹

ResistanceohmΩkg m² s⁻³ A⁻²

Magnetic flux densityteslaTkg s⁻² A⁻¹

Worked Example

Deriving the Volt04

Show that the SI base units of voltage are kg m² s⁻³ A⁻¹.

1

Start with:

V = \frac{E}{Q}

where E is energy and Q is charge

2

Energy:

E = \frac{1}{2} m v^{2}

→ units: kg × (m s⁻¹)² = kg m² s⁻²

3

Charge:

Q = I t

→ units: A × s = A s

4

Voltage:

V = \frac{kg m ^{2} s ^{- 2}}{A s} = kg m^{2} s^{- 3} A^{- 1}

Answer: 1 V = 1 kg m² s⁻³ A⁻¹

Order of Magnitude

Definition

The order of magnitude of a quantity is the power of 10 that most closely approximates its value. It describes the scale or size of a quantity.

Rounding Rule

If the coefficient is ≥ 5, round up to the next power of 10. If < 5, keep the current power.

Example: 5 × 10⁸ → 10⁹ (round up) | 3 × 10⁸ → 10⁸ (keep)

Worked Example

Order of Magnitude Calculation05

The diameter of a hydrogen atom is 1.06 × 10⁻¹⁰ m. Assuming it is perfectly spherical, find the approximate surface area to the nearest order of magnitude.

1

Surface area of sphere:

A = 4 π r^{2}

2

Radius = diameter ÷ 2 = 0.53 × 10⁻¹⁰ m

3

A = 4 π \times (0.53 \times 1 0^{- 10})^{2}

4

A = 4 π \times 0.28 \times 1 0^{- 20} = 3.5 \times 1 0^{- 20} m^{2}

5

3.5 rounds to nearest power → 10⁻²⁰ m²

Reference Values for Estimation

Memorise these standard values to anchor your estimations:

QuantityValueOrder of Mag.

Diameter of an atom~10⁻¹⁰ m−10

Diameter of a nucleus~10⁻¹⁵ m−15

Mass of an electron9.11 × 10⁻³¹ kg−30

Mass of a proton/neutron1.67 × 10⁻²⁷ kg−27

Human height~1.7 m0 (i.e. 1 m)

Human mass~70 kg2

Walking speed~1 m s⁻¹0

Speed of light3 × 10⁸ m s⁻¹8

Mass of the Sun~2 × 10³⁰ kg30

Fermi Estimation

Named after physicist Enrico Fermi, this technique breaks complex problems into simpler estimations.

Method

Identify the key quantities needed

Estimate each to one significant figure

Combine using appropriate relationships

Express final answer as an order of magnitude

Why It Works

Over-estimates and under-estimates tend to cancel out. The goal is the correct order of magnitude, not an exact answer. Estimation is used to make comparisons or check if a calculated value is reasonable.

Worked Example

Fermi Estimation06

Estimate the number of heartbeats in a human lifetime.

1

Heart rate: ~70 beats per minute ≈ 10² bpm

2

Minutes per year: 60 × 24 × 365 ≈ 5 × 10⁵ min

3

Lifetime: ~80 years ≈ 10² years

4

Total beats: 10² × 5 × 10⁵ × 10² = 5 × 10⁹

Answer: Order of magnitude 10⁹ to 10¹⁰ heartbeats

Practice

Question 1: Convert 2.5 GHz to Hz.

1

Identify powers: G = 10⁹, base unit = 10⁰

2

Calculate difference: 9 − 0 = 9

3

Multiply: 2.5 × 10⁹ = 2,500,000,000

Answer: 2.5 GHz = 2.5 × 10⁹ Hz

Practice

Question 2: Convert 45 kW h to J.

1

Convert kW to W: 45 kW = 45 × 10³ W = 45,000 W

2

Convert hours to seconds: 1 h = 3600 s

3

Energy = Power × time: 45,000 × 3600 = 1.62 × 10⁸ J

Answer: 45 kW h = 1.62 × 10⁸ J

Practice

Question 3: Estimate the number of atoms in the human body.

1

Human mass ≈ 70 kg, mostly water (H₂O)

2

Molar mass of water = 18 g mol⁻¹ = 0.018 kg mol⁻¹

3

Moles of water = 70 / 0.018 ≈ 4000 mol ≈ 4 × 10³ mol

4

Molecules = 4 × 10³ × 6 × 10²³ ≈ 2.4 × 10²⁷ molecules

5

Atoms per molecule = 3, so atoms ≈ 3 × 2.4 × 10²⁷ ≈ 7 × 10²⁷

Answer: Order of magnitude 10²⁸ atoms

Summary

Key Points

TopicKey Information

SI Base Unitsm, kg, s, A, K, mol — all other units derive from these

PrefixesT, G, M, k, c, m, μ, n, p, f — powers from 10¹² to 10⁻¹⁵

Standard FormA × 10ⁿ where 1 ≤ A < 10

Key Conversions1 eV = 1.6 × 10⁻¹⁹ J | 1 kW h = 3.6 MJ

Derived UnitsUse defining equations to express in base units

EstimationRound to 1 s.f., combine, give order of magnitude

Exam Skills

Always convert to SI base units before calculations

Check your answer makes sense using order of magnitude

Know key reference values for estimation questions

Watch out for mixed units (e.g. 2 h 20 min ≠ 2.2 h)