Physical quantities, units and scale
Key idea: A physical quantity needs a number and a unit. Prefixes express powers of ten without changing its value; scalars have magnitude only, while vectors also have direction.
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Guided lessons · coming soon
Guided lesson
Guided steps
- Read why a measurement needs both a number and a unit.
- Choose the option containing both a number and a unit.
- Read how prefixes change the unit and the number.
- Follow how to express a hair’s width in micrometres.
- Convert each value to the unit on its field.
- Convert the currents between amperes and milliamperes.
- Read why area and volume need squared and cubed factors.
- Compare converting the area with converting each side first.
- Convert the area and volume using the right powers.
- Correct the volume by cubing the length conversion factor.
- Compare familiar sizes using powers of ten in metres.
- Decide whether the proposed door height is sensible.
- Convert to metres and compare the powers of ten.
- Read why vectors need direction as well as size.
- Compare distance and displacement after one complete lap.
- Decide whether each quantity needs a direction.
- Find total distance, displacement and the type of temperature quantity.
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ContinuePhysics measures things from the width of an atom to the size of the Earth, so every measurement needs a unit and often a prefix such as milli or mega to keep the number readable. The working uses powers of ten (10³ = 1000 and 10⁻³ = 0.001) and standard form: a number written as a × 10ⁿ with a from 1 up to, but not including, 10, such as 4.5 × 10⁻⁷.
Physical quantities: a number and a unit
A physical quantity is a property you can measure, such as length, mass or time. It is written as a numerical magnitude and a unit. In “length = 2.4 m”, 2.4 is the magnitude and m (metre) is the unit.
The unit is part of the value. A rope 2.4 m long and a pencil 2.4 cm long have the same number, but the rope is 100 times longer. “The rope is 2.4” does not say how long the rope is, so a bare number is not a complete answer.
SI base quantities and units
Scientists use one system of units, the SI (International System of Units). It is built on a small set of base quantities, each with a base unit. Recall these six:
| Base quantity | Base unit | Symbol |
|---|---|---|
| mass | kilogram | kg |
| length | metre | m |
| time | second | s |
| electric current | ampere | A |
| temperature | kelvin | K |
| amount of substance | mole | mol |
The complete SI has seven base units, including candela for luminous intensity; this course requires the six above. Derived units combine base units. Speed is measured in m/s, a length divided by a time. Density is measured in kg/m³, a mass divided by a volume.
Three details catch people out:
- The base unit of mass is the kilogram, not the gram.
- Temperature is often measured in degrees Celsius (°C), but its SI base unit is the kelvin (K).
- Capital and small letters mean different things. m is the metre but M means mega; K is the kelvin but k means kilo.
Check your understanding 1
Which option gives a complete physical quantity?
Show answer
25 kg is complete: it gives both a magnitude and a unit. 25 alone has no unit; kilograms alone names a unit without an amount.
SI prefixes
Many quantities are far larger or smaller than one base unit. A prefix in front of a unit multiplies the unit by a power of ten, so the number stays easy to read. A prefix rescales the unit without changing the quantity: 3.2 km and 3200 m are the same length.
| Prefix | Symbol | Multiplies the unit by |
|---|---|---|
| tera | T | 10¹² |
| giga | G | 10⁹ |
| mega | M | 10⁶ |
| kilo | k | 10³ |
| deci | d | 10⁻¹ |
| centi | c | 10⁻² |
| milli | m | 10⁻³ |
| micro | µ | 10⁻⁶ |
| nano | n | 10⁻⁹ |
From a prefixed unit to the SI unit
Replace the prefix symbol with its power of ten, then write the answer in standard form.
| Quantity | Replace the prefix | In SI units, standard form |
|---|---|---|
| 450 nm | 450 × 10⁻⁹ m | 4.5 × 10⁻⁷ m |
| 3.5 µA | 3.5 × 10⁻⁶ A | 3.5 × 10⁻⁶ A |
| 2.5 dm | 2.5 × 10⁻¹ m | 2.5 × 10⁻¹ m (0.25 m) |
| 2.4 GHz | 2.4 × 10⁹ Hz | 2.4 × 10⁹ Hz |
| 1.5 TJ | 1.5 × 10¹² J | 1.5 × 10¹² J |
In the first row, 450 = 4.5 × 10², so 450 × 10⁻⁹ = 4.5 × 10² × 10⁻⁹ = 4.5 × 10⁻⁷: add the powers when you multiply.
From the SI unit to a prefixed unit
Divide by the prefix’s factor. Choose the prefix that gives a convenient number, usually between 1 and 1000.
Worked example 1
A hair’s width in micrometres
Problem
A human hair is 0.000 07 m wide. Write this in micrometres.
Show full solution
Find the prefix value
Method
Look up micro.Reason
1 µm = 10⁻⁶ m, so the number of micrometres is the length in metres divided by 10⁻⁶.Working
Answer
Divide by 10⁻⁶
Convert
Method
Write the length in standard form, then divide.Reason
Standard form makes the powers of ten easy to combine.Working
Answer
0.000 07 = 7 × 10⁻⁵, and 7 × 10⁻⁵ ÷ 10⁻⁶ = 70. The hair is 70 µm wide.
Check the direction
Working
Answer
A micrometre is much smaller than a metre, so the number must get much larger. It did: 0.000 07 became 70.
Worked example 2
The Earth’s radius with a prefix
Problem
The Earth’s radius is 6 400 000 m. Write it in standard form and with a prefix.
Show full solution
Write it in standard form
Method
Move the decimal point six places.Reason
Standard form shows the power of ten directly.Working
Answer
6.4 × 10⁶ m
Match the power to a prefix
Working
Answer
Mega means 10⁶, so it is 6.4 Mm, or 6400 km.
To go from one prefix to another, pass through the SI unit. For 2.5 mm in micrometres: 2.5 mm = 2.5 × 10⁻³ m, and 2.5 × 10⁻³ ÷ 10⁻⁶ = 2.5 × 10³, so 2.5 mm = 2500 µm.
A smaller unit always needs a larger number. 0.25 m is 25 cm, not 0.0025 cm.
Check your understanding 2
(a) write 36 mA in amperes; (b) write 0.000 000 52 m in nanometres; (c) write 7.5 × 10⁹ W in gigawatts; (d) write 1.2 × 10¹³ J in terajoules.
Show answer
(a) 36 × 10⁻³ A = 3.6 × 10⁻² A = 0.036 A.
(b) 0.000 000 52 = 5.2 × 10⁻⁷, and 5.2 × 10⁻⁷ ÷ 10⁻⁹ = 5.2 × 10², so 520 nm.
(c) 7.5 × 10⁹ ÷ 10⁹ = 7.5, so 7.5 GW.
(d) 1.2 × 10¹³ ÷ 10¹² = 12, so 12 TJ.
Correct the scale factor. “Milli means 1000, so 8 mm is 8000 m” applies the factor in the wrong direction. One millimetre is one thousandth of a metre: 8 mm = 8 × 0.001 m = 0.008 m. The physical length stays the same. Expressing it in the larger unit must give a smaller number; 8000 m would turn a small length into kilometres. Dividing by 100 instead would use the centimetre factor, not the millimetre factor.
Check your understanding 3
A learner writes “42 mA = 42 000 A”. Convert 42 mA to amperes, then convert 0.006 A to milliamperes. Which conversion rule corrects the first statement?
Show answer
42 mA = 42 × 0.001 A = 0.042 A. An ampere is the larger unit, so fewer amperes describe the same current. For the reverse conversion, 0.006 A ÷ 0.001 = 6 mA: more of the smaller unit is needed.
Converting area and volume units
An area unit contains two lengths and a volume unit contains three, so the length conversion is applied once for each length. Since 1 cm = 10⁻² m:
- 1 cm² = (10⁻² m)² = 10⁻⁴ m²
- 1 cm³ = (10⁻² m)³ = 10⁻⁶ m³
- 1 dm³ = (10 cm)³ = 1000 cm³, the volume also called one litre. Since 1 dm = 0.1 m, 1 dm³ = 10⁻³ m³.
Worked example 3
The area of a floor tile
Problem
A floor tile measures 20 cm by 30 cm. Find its area in square metres.
Show full solution
Find the area in cm²
Method
Multiply the sides.Reason
Area is length × width.Working
Answer
20 × 30 = 600 cm²
Convert to m²
Method
Multiply by 10⁻⁴.Reason
1 cm is 10⁻² m, so 1 cm² is (10⁻²)² = 10⁻⁴ m².Working
Answer
600 × 10⁻⁴ = 0.060 m²
Check
Working
Answer
Converting the sides first gives the same answer: 0.20 m × 0.30 m = 0.060 m². Dividing 600 by 100 instead gives 6 m², about the floor area of a small room, which cannot be one tile.
In the same way, a 250 cm³ drink can holds 250 × 10⁻⁶ m³ = 2.5 × 10⁻⁴ m³.
Check your understanding 4
Convert 1.5 m² to cm² and 2.0 × 10⁻³ m³ to cm³.
Show answer
1 m = 100 cm, so 1 m² = (100 cm)² = 10⁴ cm² and 1 m³ = (100 cm)³ = 10⁶ cm³.
1.5 m² = 1.5 × 10⁴ cm² = 15 000 cm².
2.0 × 10⁻³ m³ = 2.0 × 10⁻³ × 10⁶ cm³ = 2.0 × 10³ cm³ = 2000 cm³, or 2 litres.
Check your understanding 5
A learner converts 500 cm³ to 5 m³ by dividing by 100. Find the correct volume in m³. Which factor is needed for a volume?
Show answer
500 × 10⁻⁶ = 5.0 × 10⁻⁴ m³. The student divided by 100, the length factor, once; a volume needs it three times, so divide by 100³ = 10⁶. A 5 m³ bottle would be bigger than a wardrobe.
Orders of magnitude: from an atom to the Earth
An order of magnitude is a size rounded to the nearest power of ten. A person 1.7 m tall has a height of order 10⁰ m, which is 1 m. The Earth’s diameter, 1.3 × 10⁷ m, is of order 10⁷ m. Going up one order of magnitude means ten times larger.
| Object | Typical size | Order of magnitude |
|---|---|---|
| diameter of the Earth | 1.3 × 10⁷ m | 10⁷ m |
| height of Mount Everest | 8.8 km = 8.8 × 10³ m | 10⁴ m |
| height of a 50-storey building | 150 m = 1.5 × 10² m | 10² m |
| height of a person | 1.7 m | 10⁰ m |
| width of a hand | 9 cm = 9 × 10⁻² m | 10⁻¹ m |
| width of a fingernail | 1 cm = 10⁻² m | 10⁻² m |
| thickness of a coin | 2 mm = 2 × 10⁻³ m | 10⁻³ m |
| width of a human hair | 0.07 mm = 7 × 10⁻⁵ m | 10⁻⁴ m |
| width of a red blood cell | 8 µm = 8 × 10⁻⁶ m | 10⁻⁵ m |
| diameter of a typical atom | 0.1 nm = 10⁻¹⁰ m | 10⁻¹⁰ m |
Orders of magnitude let you compare sizes quickly. The Earth’s diameter is about 10⁷ ÷ 10⁰ = 10⁷ times a person’s height: seven orders of magnitude, or ten million times. From an atom to the Earth is 10⁷ ÷ 10⁻¹⁰ = 10¹⁷, seventeen orders of magnitude.
They also let you reject answers that cannot be right.
Worked example 4
Is the answer sensible?
Problem
A student’s calculation gives the height of a classroom door as 2 × 10⁻² m. Is this sensible?
Show full solution
Picture the answer
Method
Convert to a familiar size.Reason
A familiar object shows whether the size is plausible.Working
Answer
2 × 10⁻² m is 2 cm, the order of magnitude of a fingernail.
Estimate the real size
Method
Compare with a person.Reason
A door is a little taller than a person.Working
Answer
Order of 10⁰ m (about 2 m)
Decide
Working
Answer
The answer is 100 times too small, so the student should look for a unit mistake, such as dividing by 100 twice when converting centimetres to metres.
Check your understanding 6
Estimate each length as a power of ten in metres: a pencil about 18 cm long, a four-storey building about 12 m high, and paper about 0.1 mm thick. How many orders of magnitude separate a person (about 1 m) and an atom (about 10⁻¹⁰ m)?
Show answer
(a) A pencil is about 18 cm = 1.8 × 10⁻¹ m, so 10⁻¹ m.
(b) Four storeys of about 3 m are about 12 m, so 10¹ m.
(c) A sheet of paper is about 0.1 mm = 10⁻⁴ m thick, so 10⁻⁴ m.
A person is 10⁰ ÷ 10⁻¹⁰ = 10¹⁰ times an atom: ten orders of magnitude.
Scalars and vectors
A scalar quantity has magnitude only. A vector quantity has magnitude and direction. Both have units.
To decide, ask whether the direction changes what happens. A bag has a mass of 5 kg whichever way it is turned, so mass is a scalar. A 30 N push on a box has a different effect to the left than to the right, so “the force is 30 N” is incomplete for a force; write “30 N to the left”. Weight is a force, so it has a direction too: downwards, towards the centre of the Earth.
| Quantity | Scalar or vector | Why |
|---|---|---|
| distance | scalar | the length of the path, whichever way it goes |
| displacement | vector | the distance from the start in a stated direction |
| speed | scalar | how fast, in any direction |
| velocity | vector | speed in a stated direction, such as 12 m/s east |
| acceleration | vector | a change of velocity has a direction |
| force, including weight | vector | a push or pull acts in a direction |
| mass | scalar | the amount of matter; it has no direction |
| time, energy, temperature | scalar | none of them points anywhere |
Three pairs are easy to mix up: distance and displacement, speed and velocity, and mass and weight. In each pair the second quantity is the vector.
Worked example 5
Distance and displacement on a track
Problem
A runner completes one lap of a 400 m track and stops at the starting line. Find the distance run and the displacement.
Show full solution
Distance
Method
Add up the length of the path.Reason
Distance is a scalar: it has size only.Working
Answer
400 m
Displacement
Working
Answer
0 m: the runner ends where they started, so they are no distance from the start in any direction.
A minus sign on a scalar is not a direction. A temperature of −5 °C is below zero; it does not point anywhere, so temperature is still a scalar.
Check your understanding 7
Classify 20 N upwards, 12 m/s due north and 25 °C as scalar or vector quantities.
Show answer
20 N upwards is a vector force; 12 m/s due north is a vector velocity. Their directions are part of their values. 25 °C is a scalar temperature: it has no spatial direction.
Check your understanding 8 A returning journey and a signed temperature
A learner walks 4 m east and then 3 m west along one straight path. Find the distance travelled and signed displacement, taking east as positive. Is a room temperature of −5 °C a scalar or a vector?
Show answer
Distance adds both parts of the path: 4 + 3 = 7 m. The learner ends 1 m east of the start, so displacement is 1 m east. The −5 °C value is below the Celsius zero; it does not give a spatial direction. Temperature remains a scalar.
Common mistakes
- Giving an answer with no unit. 2.4 m and 2.4 cm are different lengths.
- Writing M for milli or m for mega. M is 10⁶ and m is 10⁻³.
- Converting in the wrong direction. A smaller unit needs a larger number: 0.25 m is 25 cm, not 0.0025 cm.
- Converting cm² or cm³ with the length factor only once. 1 cm² = 10⁻⁴ m² and 1 cm³ = 10⁻⁶ m³.
- Giving the gram as the base unit of mass. The SI base unit is the kilogram.
- Calling speed a vector or weight a scalar. Speed and mass are scalars; velocity and weight are vectors.
- Calling temperature a vector because it can be negative. A minus sign is not a direction.
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Syllabus and review details
- SEC G3 Combined Science Physics component 2027 · 2027
Physics section, PDF pages 8–24
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