Physical quantities

Q: What two components make up every physical quantity?

Every physical quantity consists of a numerical magnitude and a unit .

Q: How should compound units be formatted in CCEA examinations?

Compound units must be written in linear format using negative indices (e.g., \(\text{m s}^{-1}\)), not using slashes (e.g., \(\text{m/s}\)).

Q: What is a derived unit in physics?

A derived unit is a unit defined by combining (multiplying or dividing) two or more SI base units.

Physics 1210 · CCEA A Level · 8 min read · Updated: Aug 31, 2026

Chapter 1.1: Physical Quantities

Welcome to AS Level Physics! Whether you found GCSE physics straightforward or a bit of a challenge, this chapter lays the groundwork for everything you will study in Unit AS 1: Forces, Energy and Electricity. Physical quantities are the common language of scientists across the globe. By understanding how measurements work, how units fit together, and how to convert prefixes, you will unlock easy marks in your CCEA examinations.

Don't worry if breaking down equations into base units seems strange at first. With a few simple steps and a bit of practice, it becomes completely systematic!

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1. What is a Physical Quantity?

In physics, a physical quantity is any property of an object or system that can be measured and given a numerical value.

Every physical quantity must consist of two essential parts:
1. A numerical magnitude (how much or how many).
2. A unit (the standard of measurement).

Analogy: Imagine walking into a bakery and asking for "5". The baker will look confused: 5 loaves of bread? 5 grams of flour? 5 kilograms of dough? A number alone has no physical meaning. You must state both the number and the unit: \(5\text{ kg}\).

Quick Review:
Physical Quantity \(=\) Magnitude \(\times\) Unit
For example, in a speed of \(12\text{ m s}^{-1}\), \(12\) is the numerical magnitude and \(\text{m s}^{-1}\) is the unit.

Key Takeaway: Never write a number down in a physics calculation without checking if it needs a unit. In CCEA exam papers, missing units are an easy way to lose marks!

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2. The SI System and Base Units

To ensure scientists around the world understand each other, the scientific community uses the Système International d'Unités (the SI system). This system is built upon a small set of fundamental building blocks called base quantities and their corresponding SI base units.

For your CCEA AS 1 specification, there are six primary base quantities and units that you must know:

• Mass: base unit is the kilogram, symbol \(\text{kg}\)
• Length: base unit is the metre, symbol \(\text{m}\)
• Time: base unit is the second, symbol \(\text{s}\)
• Electric Current: base unit is the ampere, symbol \(\text{A}\)
• Temperature (Thermodynamic): base unit is the kelvin, symbol \(\text{K}\)
• Amount of Substance: base unit is the mole, symbol \(\text{mol}\)

Note: The seventh SI base unit is luminous intensity (candela, \(\text{cd}\)), but it is not typically tested computationally at AS level.

Crucial CCEA Examination Rule: Linear Index Notation

CCEA examiners require compound units to be written in linear format using negative indices rather than using slashes (solidi).
• Write metres per second as \(\text{m s}^{-1}\) (NOT \(\text{m/s}\))
• Write kilograms per cubic metre as \(\text{kg m}^{-3}\) (NOT \(\text{kg/m}^3\))

Key Takeaway: All other physical units in physics are created by combining these base units together.

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3. Derived Quantities and Units

A derived unit is a unit defined by combining (multiplying or dividing) two or more SI base units.

Many derived units have special names (like the Newton for force, or the Joule for energy), but they can always be broken down into fundamental SI base units. CCEA AS 1 exam papers frequently ask you to express a derived unit in terms of its SI base units. Here is the step-by-step method to do this every time:

Step-by-Step Method for Deriving Base Units

Step 1: Write down a defining formula for the quantity that you know well.
Step 2: Replace each quantity in the formula with its fundamental units.
Step 3: Simplify the indices and write the final answer in linear format.

Core Derivations for AS 1:

1. Velocity (\(v\))
• Formula: \(v = \frac{d}{t}\)
• Units: \(\frac{\text{m}}{\text{s}}\)
• SI Base Units: \(\text{m s}^{-1}\)

2. Acceleration (\(a\))
• Formula: \(a = \frac{\Delta v}{t}\)
• Units: \(\frac{\text{m s}^{-1}}{\text{s}}\)
• SI Base Units: \(\text{m s}^{-2}\)

3. Force (\(F\))
• Derived Unit: Newton (\(\text{N}\))
• Formula: \(F = ma\)
• Units: \(\text{kg} \times \text{m s}^{-2}\)
• SI Base Units: \(\text{kg m s}^{-2}\)

4. Work Done or Energy (\(W\) or \(E\))
• Derived Unit: Joule (\(\text{J}\))
• Formula: \(W = F \times s\) (Force \(\times\) distance)
• Units: \((\text{kg m s}^{-2}) \times \text{m}\)
• SI Base Units: \(\text{kg m}^2 \text{s}^{-2}\)

5. Power (\(P\))
• Derived Unit: Watt (\(\text{W}\))
• Formula: \(P = \frac{W}{t}\) (Work done \(\div\) time)
• Units: \(\frac{\text{kg m}^2 \text{s}^{-2}}{\text{s}}\)
• SI Base Units: \(\text{kg m}^2 \text{s}^{-3}\)

6. Pressure (\(p\))
• Derived Unit: Pascal (\(\text{Pa}\))
• Formula: \(p = \frac{F}{A}\) (Force \(\div\) area)
• Units: \(\frac{\text{kg m s}^{-2}}{\text{m}^2}\)
• SI Base Units: \(\text{kg m}^{-1} \text{s}^{-2}\)

7. Electric Charge (\(Q\))
• Derived Unit: Coulomb (\(\text{C}\))
• Formula: \(Q = I \times t\) (Current \(\times\) time)
• SI Base Units: \(\text{A s}\)

8. Potential Difference or EMF (\(V\))
• Derived Unit: Volt (\(\text{V}\))
• Formula: \(V = \frac{W}{Q}\) (Work done \(\div\) charge)
• Units: \(\frac{\text{kg m}^2 \text{s}^{-2}}{\text{A s}}\)
• SI Base Units: \(\text{kg m}^2 \text{s}^{-3} \text{A}^{-1}\)

9. Electrical Resistance (\(R\))
• Derived Unit: Ohm (\(\Omega\))
• Formula: \(R = \frac{V}{I}\) (Potential difference \(\div\) current)
• Units: \(\frac{\text{kg m}^2 \text{s}^{-3} \text{A}^{-1}}{\text{A}}\)
• SI Base Units: \(\text{kg m}^2 \text{s}^{-3} \text{A}^{-2}\)

Key Takeaway: Never stop halfway in an exam question asking for base units! Stating that power is \(\text{J s}^{-1}\) or \(\text{N m s}^{-1}\) will not gain full credit because Joules (\(\text{J}\)) and Newtons (\(\text{N}\)) are derived units, not base units.

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4. Principle of Dimensional Homogeneity

In physics, you cannot add apples to oranges. In the same way, you cannot add a velocity to a force, or equate an energy to a power.

The Principle of Dimensional Homogeneity states that for any physical equation to be valid, every term added, subtracted, or on either side of the equals sign must have the exact same SI base units.

Important Rules for Checking Homogeneity:

• Pure numbers and dimensionless constants: Fractions like \(\frac{1}{2}\) or constants like \(\pi\) and \(2\pi\) have no units and can be ignored during base unit checks.
• Add / Subtract: If an equation reads \(s = ut + \frac{1}{2}at^2\), then \(s\), \(ut\), and \(\frac{1}{2}at^2\) must each independently have the base unit of metres (\(\text{m}\)).

Worked Example: Checking an Equation

Show that the kinematic equation \(s = ut + \frac{1}{2}at^2\) is homogeneous with respect to base units.

• Left-hand side (\(s\)): Base unit \(= \text{m}\)
• First term on right-hand side (\(ut\)): \((\text{m s}^{-1}) \times (\text{s}) = \text{m}\)
• Second term on right-hand side (\(\frac{1}{2}at^2\)): \(\frac{1}{2}\) has no units. Units \(= (\text{m s}^{-2}) \times (\text{s}^2) = \text{m}\)
• Conclusion: Every term has the SI base unit \(\text{m}\). Therefore, the equation is homogeneous.

Key Takeaway: Homogeneity confirms that an equation is dimensionally possible, but remember that it cannot tell you if a dimensionless constant (like \(\frac{1}{2}\)) is missing or incorrect!

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5. Standard SI Prefixes and Conversions

Physical quantities can range from the subatomic to the astronomical. Instead of writing extremely large or small numbers with dozens of zeros, we use standard SI prefixes.

You are required to recall and use the following 10 metric prefixes for CCEA AS 1:

• Tera- (\(\text{T}\)): \(\times 10^{12}\) (e.g., \(1\text{ THz} = 1 \times 10^{12}\text{ Hz}\))
• Giga- (\(\text{G}\)): \(\times 10^9\) (e.g., \(1\text{ GW} = 1 \times 10^9\text{ W}\))
• Mega- (\(\text{M}\)): \(\times 10^6\) (e.g., \(1\text{ MJ} = 1 \times 10^6\text{ J}\))
• kilo- (\(\text{k}\)): \(\times 10^3\) (e.g., \(1\text{ km} = 1 \times 10^3\text{ m}\))
• centi- (\(\text{c}\)): \(\times 10^{-2}\) (e.g., \(1\text{ cm} = 1 \times 10^{-2}\text{ m}\))
• milli- (\(\text{m}\)): \(\times 10^{-3}\) (e.g., \(1\text{ mA} = 1 \times 10^{-3}\text{ A}\))
• micro- (\(\mu\)): \(\times 10^{-6}\) (e.g., \(1\text{ }\mu\text{m} = 1 \times 10^{-6}\text{ m}\))
• nano- (\(\text{n}\)): \(\times 10^{-9}\) (e.g., \(1\text{ nm} = 1 \times 10^{-9}\text{ m}\))
• pico- (\(\text{p}\)): \(\times 10^{-12}\) (e.g., \(1\text{ pF} = 1 \times 10^{-12}\text{ F}\))
• femto- (\(\text{f}\)): \(\times 10^{-15}\) (e.g., \(1\text{ fm} = 1 \times 10^{-15}\text{ m}\))

Mastering Area and Volume Conversions (Examiner Trap!)

Converting linear units with prefixes is straightforward, but converting areas and volumes is one of the most common places students drop marks. Remember to apply the power to the prefix conversion factor:

Area Conversion (\(\text{cm}^2\) to \(\text{m}^2\)):
\(1\text{ cm} = 10^{-2}\text{ m}\)
\(1\text{ cm}^2 = (10^{-2}\text{ m})^2 = 10^{-4}\text{ m}^2\)
Multiply by \(10^{-4}\), not \(10^{-2}\)!

Area Conversion (\(\text{mm}^2\) to \(\text{m}^2\)):
\(1\text{ mm} = 10^{-3}\text{ m}\)
\(1\text{ mm}^2 = (10^{-3}\text{ m})^2 = 10^{-6}\text{ m}^2\)
Multiply by \(10^{-6}\), not \(10^{-3}\)!

Volume Conversion (\(\text{mm}^3\) to \(\text{m}^3\)):
\(1\text{ mm} = 10^{-3}\text{ m}\)
\(1\text{ mm}^3 = (10^{-3}\text{ m})^3 = 10^{-9}\text{ m}^3\)
Multiply by \(10^{-9}\), not \(10^{-3}\)!

Key Takeaway: Whenever you see an area in \(\text{mm}^2\) or a volume in \(\text{cm}^3\), convert it to standard SI base units (\(\text{m}^2\) or \(\text{m}^3\)) before substituting it into any formula.

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6. Common Exam Pitfalls & How to Avoid Them

• Pitfall 1: Case Sensitivity Errors.
Capitalisation matters! Writing lowercase \(\text{m}\) instead of capital \(\text{M}\) changes your answer by a factor of \(10^9\) (milli- is \(10^{-3}\), Mega- is \(10^6\)). Similarly, do not mix up lowercase \(\text{k}\) (kilo) with capital \(\text{K}\) (kelvin).

• Pitfall 2: Using Slashes in Final Units.
Always use index notation. Write \(\text{kg m s}^{-2}\), never \(\text{kg m/s}^2\).

• Pitfall 3: Incomplete Base Unit Reductions.
If asked for SI base units of potential difference, writing \(\text{J C}^{-1}\) gets \(0\) marks. You must break both Joules and Coulombs down to \(\text{kg m}^2 \text{s}^{-3} \text{A}^{-1}\).

• Pitfall 4: Forgetting that Constants are Dimensionless.
In equations like \(E_k = \frac{1}{2}mv^2\), ignore the \(\frac{1}{2}\) completely when deriving or checking base units.

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Quick Summary Checklist

Before sitting your AS 1 exam, make sure you can confidently:
1. Identify all 6 key SI base units (\(\text{kg}\), \(\text{m}\), \(\text{s}\), \(\text{A}\), \(\text{K}\), \(\text{mol}\)).
2. Derive the base units for force, energy, power, pressure, charge, voltage, and resistance.
3. Prove that an equation is homogeneous by showing both sides simplify to the same base units.
4. Convert between metric prefixes from \(\text{Tera-} (10^{12})\) down to \(\text{femto-} (10^{-15})\).
5. Correctly convert area and volume units (e.g., \(\text{cm}^2 \to 10^{-4}\text{ m}^2\)).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What two components make up every physical quantity?

Every physical quantity consists of a numerical magnitude and a unit.

What are the six primary SI base quantities and their corresponding base units for AS Physics?

Mass (\(\text{kg}\)), Length (\(\text{m}\)), Time (\(\text{s}\)), Electric Current (\(\text{A}\)), Thermodynamic Temperature (\(\text{K}\)), and Amount of Substance (\(\text{mol}\)).

How should compound units be formatted in CCEA examinations?

Compound units must be written in linear format using negative indices (e.g., \(\text{m s}^{-1}\)), not using slashes (e.g., \(\text{m/s}\)).

What is a derived unit in physics?

A derived unit is a unit defined by combining (multiplying or dividing) two or more SI base units.

What are the SI base units for Force, Energy, and Power?

Force is \(\text{kg m s}^{-2}\), Energy is \(\text{kg m}^2 \text{s}^{-2}\), and Power is \(\text{kg m}^2 \text{s}^{-3}\).

What are the SI base units for Electric Charge, Potential Difference, and Resistance?

Charge is \(\text{A s}\), Potential Difference is \(\text{kg m}^2 \text{s}^{-3} \text{A}^{-1}\), and Resistance is \(\text{kg m}^2 \text{s}^{-3} \text{A}^{-2}\).

What is the Principle of Dimensional Homogeneity?

For an equation to be valid, every term added, subtracted, or on either side of the equals sign must have the exact same SI base units.

What multiplier is represented by each of the SI prefixes from femto- (\(\text{f}\)) to Tera- (\(\text{T}\))?

\(\text{f}=10^{-15}\), \(\text{p}=10^{-12}\), \(\text{n}=10^{-9}\), \(\mu=10^{-6}\), \(\text{m}=10^{-3}\), \(\text{c}=10^{-2}\), \(\text{k}=10^3\), \(\text{M}=10^6\), \(\text{G}=10^9\), and \(\text{T}=10^{12}\).

How do you correctly convert area and volume units involving prefixes, such as \(\text{cm}^2\) or \(\text{mm}^3\), to standard SI units (\(\text{m}^2\) or \(\text{m}^3\))?

Raise the linear conversion factor to the power of the dimension: multiply \(\text{cm}^2\) by \(10^{-4}\text{ m}^2\), \(\text{mm}^2\) by \(10^{-6}\text{ m}^2\), and \(\text{mm}^3\) by \(10^{-9}\text{ m}^3\).

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