GCSE · Physics · Edexcel · Spec 1PH0

Converting between units

Is 72 km/h fast? In metres per second it's a tidy 20. Same speed, different number. Converting units changes the number, never the thing.

One distance, two numbers

Drag the walk. Watch the numbers.

the distance walked: 5/10. Distance in kilometres 2.5 km. The same distance in metres 2500 m. km → m: multiply by 1000 2.5 × 1000 = 2500. m → km: divide by 1000 2500 ÷ 1000 = 2.5

Distance in kilometres2.5 kmThe same distance in metres2500 mkm → m: multiply by 10002.5 × 1000 = 2500m → km: divide by 10002500 ÷ 1000 = 2.5

A walk of up to 5 km. Drag along it. The distance in the two readouts is always exactly the same, but the number in metres is always 1000 times bigger than the number in kilometres.

Watch out: Kilometres to metres makes the number bigger, so you multiply. If your answer in metres comes out smaller than the number of kilometres you started with, you have divided by mistake.

The prefix ladder

Tap each prefix to see its power of ten. The base unit, with no prefix, sits in the middle at 1.

nano (n)

× 10⁻⁹

Nano means × 10⁻⁹. So 1 nm = 10⁻⁹ m = 0.000 000 001 m: a billionth of a metre.

Before you substitute

Straight in, or convert first?

Pick a value, then pick where it goes.

Still to sort

Already SI: substitute as it is (0)

The SI unit for that quantity: metres, kilograms, seconds, joules, watts, amps.

Where the line is: Ask two things: is this the SI unit for that quantity, and is there a prefix in front of it that isn't part of that unit?

Convert first (0)

A prefix that isn't part of the SI unit, or an everyday unit that isn't the SI one.

Where the line is: A unit you use every day can still be the wrong unit for the equation.

9 of 9 still to sort.

Each value is about to go into an equation that works in SI units. Decide which ones can go in as they are.

Compound units

72 km/h in metres per second

A car travels at 72 km/h. What is its speed in m/s?

km/h means kilometres per hour. There are two units to change, so take them one at a time: first the kilometres, then the hours.
missing step

Which line is step 2?

Kilometres to metres: 72 km = 72 × 1000 = 72 000 m. So the car covers 72 000 m every hour.Kilometres to metres: 72 km = 72 ÷ 1000 = 0.072 m. So the car covers 0.072 m every hour.Kilometres to metres: 72 km = 72 × 100 = 7200 m. So the car covers 7200 m every hour.

Area and volume

How many square centimetres in a square metre?

Picture a square floor tile exactly 1 m long and 1 m wide. Its area is 1 m². Now imagine covering it in tiny squares, each 1 cm by 1 cm.

How many of those 1 cm² squares fit on the tile?

100. There are 100 cm in a metre, so there are 100 cm² in a square metre.1000. Metres always convert with a factor of 1000.10 000. The tile is 100 cm by 100 cm, so it holds 100 × 100 squares.1 000 000. You multiply 100 by itself for each dimension: 100 × 100 × 100.

How sure are you?

I'm sureI think soI'm guessing

100. There are 100 cm in a metre, so there are 100 cm² in a square metre.

This is a common misconception

The 100 is right for a length. But count the squares: along one edge of the tile there are 100 of them, and there are 100 rows like that. That's 100 × 100 = 10 000 squares. Area multiplies two lengths together, so the factor of 100 gets used twice. 100 cm² would only cover one thin strip along the edge.

1000. Metres always convert with a factor of 1000.

This is a common misconception

1000 is the kilo factor (1 km = 1000 m), and it is also the factor between metres and millimetres. Between metres and centimetres it is 100, because centi means × 10⁻². Then, because this is an area, that 100 has to be used twice: 100 × 100 = 10 000.

10 000. The tile is 100 cm by 100 cm, so it holds 100 × 100 squares.

This matches the evidence

Yes. 100 squares along each edge, 100 rows: 100 × 100 = 10⁴ = 10 000 cm². Going the other way, from cm² to m², you divide by 10 000. The same idea works for volume: a 1 m cube is 100 cm along each of its three edges, so 1 m³ = 100 × 100 × 100 = 10⁶ cm³.

1 000 000. You multiply 100 by itself for each dimension: 100 × 100 × 100.

This is a common misconception

That's the right number for a cube, not a square. A 1 m cube is 100 cm in three directions, so 1 m³ = 100 × 100 × 100 = 10⁶ cm³. A flat tile only has two directions, length and width, so it's 100 × 100 = 10⁴ cm². Count the dimensions: squared means two, cubed means three.

Count the dimensions. Length: × 100 once. Area (m²): × 100 twice, so 1 m² = 10⁴ cm². Volume (m³): × 100 three times, so 1 m³ = 10⁶ cm³.

Spot the slip

Where does this answer go wrong?

A toy car travels 600 mm in 1.5 s. Find its speed in m/s.

A student's answer — which line goes wrong?

Speed in m/s means metres travelled per second, so the distance must be in metres and the time in seconds.
The time is already in seconds: 1.5 s.
Distance: 600 mm = 600 × 1000 = 600 000 m
Speed = 600 000 ÷ 1.5
Speed = 400 000 m/s

Exam line: Convert every value to its SI unit before you substitute, then check the size: a bigger unit means a smaller number.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Same quantity, different number. How to get every value into the right units before it goes into an equation, and how to tell at a glance if you've got it wrong.

What you need to know

Physics calculations need consistent units, usually SI units (for example metres, kilograms, seconds, joules, watts). Convert any other unit before you substitute into an equation.
Prefixes multiply a unit by a power of ten: giga (G) 10⁹, mega (M) 10⁶, kilo (k) 10³, centi (c) 10⁻², milli (m) 10⁻³, micro (μ) 10⁻⁶, nano (n) 10⁻⁹.
Prefixed unit to base unit: multiply by the prefix's power of ten (2.5 kW = 2500 W). Base unit to prefixed unit: divide by it (0.004 m = 4 mm).
The SI unit of mass is the kilogram, so grams are converted using 1 kg = 1000 g.
Time: 1 minute = 60 s and 1 hour = 3600 s. Hours to seconds, multiply by 3600; seconds to hours, divide by 3600.
Compound units are converted one part at a time: km/h to m/s is × 1000 then ÷ 3600, so 72 km/h = 20 m/s.
Area and volume use the square and cube of the length factor: 1 m² = 10⁴ cm² and 1 m³ = 10⁶ cm³.

The big picture

Physics equations only work when every value is in consistent units, usually SI units such as metres, kilograms, seconds, joules and watts, so anything else is converted before it is substituted. Each prefix multiplies a unit by a power of ten: to reach the base unit, multiply by it; to go back, divide. Mass goes into kilograms (1 kg = 1000 g), time into seconds (1 h = 3600 s), and a compound unit such as km/h is converted one part at a time. Area and volume use the square and cube of the length factor. Finally, check the size: a smaller unit needs a bigger number.

Key points

1Changing the unit changes the number, never the quantity.

2A smaller unit needs a larger number, and a larger unit a smaller number. Use this to check every conversion.

3The kilogram is the one SI unit that already carries a prefix; grams are not SI.

4In front of a unit, small m means milli (× 10⁻³) and capital M means mega (× 10⁶).

5For very large or very small results, write the answer in standard form, for example 5 MJ = 5 × 10⁶ J.

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Worked example

Problem

Write 2.4 GJ in joules and 480 μm in metres. Give both answers in standard form.

⚠ Watch out

Using the length factor unchanged for area or volume, such as writing 1 m² = 100 cm². The factor of 100 has to be squared for area (1 m² = 10⁴ cm²) and cubed for volume (1 m³ = 10⁶ cm³).

🧠

Memory hook

Smaller unit, bigger number. Shrink the unit and the number has to grow to describe the same thing, just as a distance in metres is always 1000 times the number in kilometres.

✓

Check yourself

Cover the page. Which of 3.5 kg, 250 g and 45 minutes need converting before they go into an equation? Convert those, and say why each new number got bigger or smaller.

Flashcards

(14)

Why must values be converted before substituting into a physics equation?

The equation only works if the units are consistent, usually SI units such as metres, kilograms, seconds, joules and watts.

What does the prefix giga (G) mean?

× 10⁹. For example, 1 GJ = 10⁹ J.

What does the prefix mega (M) mean?

× 10⁶. For example, 5 MJ = 5 × 10⁶ J.

What does the prefix kilo (k) mean?

× 10³. For example, 1 km = 1000 m.

What does the prefix centi (c) mean?

× 10⁻². For example, 1 cm = 0.01 m, so 100 cm = 1 m.

What does the prefix milli (m) mean?

× 10⁻³. For example, 1 mA = 0.001 A.

Which prefix means a millionth, × 10⁻⁶, and what is its symbol?

Micro, written μ. So 1 μm is a millionth of a metre.

Write 1 nm in metres.

Nano (n) means × 10⁻⁹, so 1 nm = 10⁻⁹ m = 0.000 000 001 m.

Going from a prefixed unit to the base unit: multiply or divide by the prefix's power of ten?

Multiply (2.5 kW = 2.5 × 10³ W = 2500 W). Going back from the base unit to the prefixed unit, divide.

What is the SI unit of mass, and how do you convert grams to it?

The kilogram. 1 kg = 1000 g, so divide a mass in grams by 1000.

How many seconds are in 1 hour?

3600 s (60 minutes × 60 s). Hours to seconds: × 3600. Seconds to hours: ÷ 3600.

How do you convert a speed from km/h to m/s?

One part at a time: × 1000 to change km to m, then ÷ 3600 to change per hour to per second.

How do the unit factors change for area and volume?

Square the length factor for area and cube it for volume: 1 m² = 100 × 100 = 10⁴ cm², and 1 m³ = 100 × 100 × 100 = 10⁶ cm³.

How can you check a unit conversion without redoing it?

Look at the size. A smaller unit needs a larger number; a larger unit needs a smaller number.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Straight in, or convert first?

72 km/h in metres per second

How many square centimetres in a square metre?

Where does this answer go wrong?

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