Why Unit Conversions Matter

Measurements are only useful when their units are understood.

In physics, the same quantity can often be expressed using different units. For example:

1.5 km = 1500 m

250 cm = 2.50 m

500 ms = 0.500 s

The physical quantity has not changed. Only the way it is expressed has changed.

Being able to convert units correctly is essential because physics equations usually require quantities to be expressed in compatible units.

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SI Units

The International System of Units (SI) provides standard units used by scientists around the world.

Some important SI units are:

Many other physics units are derived from these units.

For example:

speed → m/s

acceleration → m/s²

force → N

energy → J

pressure → Pa

density → kg/m³

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Metric Prefixes

Metric prefixes allow us to represent very large or very small quantities without writing long strings of zeros.

The prefix tells us the multiplier applied to the basic unit.

For example:

1 km = 10³ m

1 mm = 10⁻³ m

1 µm = 10⁻⁶ m

1 nm = 10⁻⁹ m

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Prefix Symbols Matter

Capital and lowercase letters can have different meanings.

For example:

m = metre

but:

mm = millimetre

and:

Mm = megametre

Similarly:

mW = milliwatt

while:

MW = megawatt

These differ by a factor of:

10⁹

because:

milli = 10⁻³

and:

mega = 10⁶

Always pay attention to capitalization.

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Converting Kilometres to Metres

Since:

1 km = 1000 m

to convert kilometres to metres, multiply by 1000.

Example

Convert:

4.2 km → m

Calculation:

4.2 × 1000 = 4200

Therefore:

4.2 km = 4200 m

or in scientific notation:

4.2 × 10³ m

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Converting Metres to Kilometres

To go in the opposite direction, divide by 1000.

Example

Convert:

7500 m → km

Calculation:

7500 ÷ 1000 = 7.5

Therefore:

7500 m = 7.5 km

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A Conversion-Factor Method

A powerful method is to multiply by a fraction equal to 1.

Suppose we want to convert:

3.6 km → m

We know:

1 km = 1000 m

Write:

3.6 km × (1000 m / 1 km)

The kilometres cancel:

3.6 × 1000 m

Therefore:

3.6 km = 3600 m

This method becomes particularly useful for more complicated conversions.

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Units Should Cancel

Treat units almost like algebraic quantities.

For example:

25 cm × (1 m / 100 cm)

The cm units cancel.

This leaves:

25/100 m

Therefore:

25 cm = 0.25 m

If the unwanted unit does not cancel, the conversion factor may be upside down.

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Converting Centimetres

Remember:

1 m = 100 cm

Therefore:

1 cm = 0.01 m

or:

1 cm = 10⁻² m

Example

Convert:

350 cm → m

350 cm × (1 m / 100 cm)

= 3.50 m

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Converting Millimetres

Remember:

1 m = 1000 mm

Therefore:

1 mm = 10⁻³ m

Example

Convert:

48 mm → m

48 mm × (1 m / 1000 mm)

= 0.048 m

or:

4.8 × 10⁻² m

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Converting Micrometres

A micrometre is:

1 µm = 10⁻⁶ m

Therefore:

25 µm = 25 × 10⁻⁶ m

Convert to standard scientific notation:

25 µm = 2.5 × 10⁻⁵ m

Micrometres are commonly used for:

• Cells.
• Microorganisms.
• Small structures.
• Thin materials.

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Converting Nanometres

Remember:

1 nm = 10⁻⁹ m

Example

Convert:

620 nm → m

620 × 10⁻⁹ m

Rewrite in scientific notation:

6.20 × 10⁻⁷ m

Therefore:

620 nm = 6.20 × 10⁻⁷ m

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Converting Mass

Mass requires particular care because the SI base unit is the kilogram, not the gram.

Remember:

1 kg = 1000 g

1 g = 1000 mg

Therefore:

1 kg = 1,000,000 mg

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Kilograms and Grams

Example

Convert:

2.75 kg → g

2.75 × 1000 = 2750

Therefore:

2.75 kg = 2750 g

Reverse Conversion

Convert:

450 g → kg

450 ÷ 1000 = 0.450

Therefore:

450 g = 0.450 kg

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Grams and Milligrams

Since:

1 g = 1000 mg

then:

3.2 g = 3200 mg

and:

850 mg = 0.850 g

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Milligrams to Kilograms

Sometimes several conversion steps are required.

Convert:

2500 mg → kg

First:

2500 mg = 2.5 g

Then:

2.5 g = 0.0025 kg

Therefore:

2500 mg = 2.5 × 10⁻³ kg

We could also do this in one step:

1 mg = 10⁻⁶ kg

so:

2500 mg = 2500 × 10⁻⁶ kg

= 2.5 × 10⁻³ kg

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Converting Time

Time is slightly different because it is not entirely based on powers of ten.

Remember:

1 min = 60 s

1 h = 60 min

Therefore:

1 h = 3600 s

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Minutes to Seconds

Convert:

4.5 min → s

4.5 × 60 = 270

Therefore:

4.5 min = 270 s

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Hours to Seconds

Convert:

2.0 h → s

We can perform two conversions:

2.0 h × (60 min / 1 h) × (60 s / 1 min)

The hours and minutes cancel.

2.0 × 60 × 60 = 7200

Therefore:

2.0 h = 7200 s

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Seconds to Hours

Convert:

18,000 s → h

Since:

1 h = 3600 s

then:

18,000 ÷ 3600 = 5

Therefore:

18,000 s = 5 h

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Milliseconds

The prefix milli means:

10⁻³

Therefore:

1 ms = 10⁻³ s

Example

Convert:

250 ms → s

250 × 10⁻³ = 0.250

Therefore:

250 ms = 0.250 s

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Microseconds and Nanoseconds

Remember:

1 µs = 10⁻⁶ s

1 ns = 10⁻⁹ s

For example:

45 µs = 45 × 10⁻⁶ s

= 4.5 × 10⁻⁵ s

And:

8.0 ns = 8.0 × 10⁻⁹ s

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Multi-Step Conversions

Some conversions require more than one conversion factor.

For example:

Convert:

72 km/h → m/s

Start with:

72 km/h

Convert kilometres to metres:

72 km/h × (1000 m / 1 km)

Then convert hours to seconds:

× (1 h / 3600 s)

Units cancel:

72 × 1000 / 3600

= 20

Therefore:

72 km/h = 20 m/s

This conversion is extremely common in physics.

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Converting m/s to km/h

Now convert:

15 m/s → km/h

Use:

15 m/s × (1 km / 1000 m) × (3600 s / 1 h)

Units cancel:

15 × 3600 / 1000

= 54

Therefore:

15 m/s = 54 km/h

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A Useful Shortcut for Speed

Because:

1 m/s = 3.6 km/h

you can use:

m/s → km/h

multiply by 3.6

km/h → m/s

divide by 3.6

For example:

90 km/h ÷ 3.6 = 25 m/s

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Why Physics Often Requires SI Units

Consider:

v = d/t

Suppose:

d = 1500 m

and:

t = 2 min

If you substitute directly:

v = 1500 / 2

you get:

750

But this is 750 m/min, not m/s.

Convert time first:

2 min = 120 s

Then:

v = 1500 / 120

v = 12.5 m/s

Always check the units before substituting values into an equation.

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Worked Physics Example: Speed

A runner travels:

5.0 km

in:

20 min

Find the average speed in m/s.

Convert distance:

5.0 km = 5000 m

Convert time:

20 min = 1200 s

Use:

v = d/t

v = 5000 / 1200

v ≈ 4.17 m/s

The unit conversions must happen before the final calculation if the answer is required in SI units.

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Worked Physics Example: Acceleration

A car changes velocity from:

36 km/h

to:

72 km/h

in:

5.0 s

Convert velocities:

36 km/h = 10 m/s

72 km/h = 20 m/s

Velocity change:

Δv = 20 − 10

Δv = 10 m/s

Use:

a = Δv/t

a = 10 / 5.0

a = 2.0 m/s²

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Worked Physics Example: Force

A cart has a mass of:

750 g

and accelerates at:

4.0 m/s²

Calculate the resultant force.

The SI unit of mass is kilograms.

Convert:

750 g = 0.750 kg

Use:

F = ma

F = 0.750 × 4.0

F = 3.0 N

If you accidentally used 750 instead of 0.750, the answer would be 1000 times too large.

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Worked Physics Example: Density

An object has:

mass = 250 g

volume = 100 cm³

Suppose the density is required in:

g/cm³

No conversion is necessary because the units are already compatible.

Use:

ρ = m/V

ρ = 250 / 100

ρ = 2.5 g/cm³

This is an important point:

Do not convert units unnecessarily.

Convert them when the equation or requested answer requires it.

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Area Conversions

Area conversions require special care because the conversion factor must also be squared.

We know:

1 m = 100 cm

Therefore:

1 m² = (100 cm)²

1 m² = 10,000 cm²

Not:

1 m² = 100 cm²

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Worked Area Example

Convert:

2.5 m² → cm²

Since:

1 m² = 10,000 cm²

then:

2.5 × 10,000 = 25,000

Therefore:

2.5 m² = 25,000 cm²

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Using Conversion Factors with Area

The same result can be shown using unit cancellation:

2.5 m² × (100 cm / 1 m)²

Square the entire conversion factor:

2.5 × 100² cm²

= 25,000 cm²

Both the number and the unit relationship must be squared.

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Volume Conversions

Volume requires even more care because the conversion factor must be cubed.

Since:

1 m = 100 cm

then:

1 m³ = (100 cm)³

1 m³ = 1,000,000 cm³

Therefore:

1 m³ = 10⁶ cm³

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Cubic Centimetres and Millilitres

A particularly useful relationship is:

1 cm³ = 1 mL

Therefore:

250 cm³ = 250 mL

Also:

1000 cm³ = 1000 mL = 1 L

So:

1 L = 1000 cm³

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Litres and Cubic Metres

Remember:

1 m³ = 1000 L

Therefore:

1 L = 0.001 m³

or:

1 L = 10⁻³ m³

And:

1 mL = 10⁻⁶ m³

These conversions are particularly useful in density, pressure, fluids, and chemistry.

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Worked Volume Example

Convert:

2.5 L → m³

Since:

1 L = 10⁻³ m³

then:

2.5 L = 2.5 × 10⁻³ m³

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Worked Volume Example: cm³ to m³

Convert:

350 cm³ → m³

Since:

1 cm = 10⁻² m

cube both sides:

1 cm³ = (10⁻²)³ m³

1 cm³ = 10⁻⁶ m³

Therefore:

350 cm³ = 350 × 10⁻⁶ m³

= 3.50 × 10⁻⁴ m³

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Prefixes with Squared and Cubed Units

This is one of the most common sources of unit-conversion errors.

Length

1 cm = 10⁻² m

Area

1 cm² = 10⁻⁴ m²

because:

(10⁻²)² = 10⁻⁴

Volume

1 cm³ = 10⁻⁶ m³

because:

(10⁻²)³ = 10⁻⁶

The exponent applies to the conversion factor as well as the unit.

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Density Conversions

Density commonly appears in:

kg/m³

or:

g/cm³

A useful relationship is:

1 g/cm³ = 1000 kg/m³

For example:

2.7 g/cm³ = 2700 kg/m³

This is approximately the density of aluminium.

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Why the Density Conversion Works

Consider:

1 g/cm³

Convert grams:

1 g = 10⁻³ kg

Convert cubic centimetres:

1 cm³ = 10⁻⁶ m³

Therefore:

1 g/cm³ = (10⁻³ kg)/(10⁻⁶ m³)

Subtract exponents:

10⁻³ / 10⁻⁶ = 10³

Therefore:

1 g/cm³ = 10³ kg/m³

or:

1 g/cm³ = 1000 kg/m³

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Compound Units

Physics frequently uses units made from several other units.

Examples include:

m/s

m/s²

kg/m³

N/m²

J/s

Each part of the compound unit must be treated correctly during conversion.

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Worked Example: Acceleration Units

Suppose acceleration is:

5.0 km/s²

Convert kilometres to metres:

5.0 km/s² × (1000 m / 1 km)

Therefore:

5.0 km/s² = 5000 m/s²

or:

5.0 × 10³ m/s²

The seconds were already in the correct unit, so they did not need conversion.

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Dimensional Checking

Units can help determine whether a calculation makes sense.

Consider:

v = d/t

If:

d → metres

and:

t → seconds

then:

v → metres/seconds

or:

m/s

This matches the expected SI unit for speed.

If your final unit were:

m·s

something would be wrong with the calculation.

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Checking Whether an Answer Is Reasonable

After converting a value, ask:

Should the numerical value become larger or smaller?

For example:

Convert:

2 km → m

A metre is a smaller unit than a kilometre.

Therefore, we need more metres to describe the same distance.

So the numerical value should increase:

2 km = 2000 m

If you obtained:

0.002 m

you should immediately recognize that the answer is unreasonable.

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Large Unit to Small Unit

When converting from a larger unit to a smaller unit, the numerical value usually increases.

For example:

3 m = 300 cm

5 kg = 5000 g

2 s = 2000 ms

The physical quantity remains unchanged.

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Small Unit to Large Unit

When converting from a smaller unit to a larger unit, the numerical value usually decreases.

For example:

5000 m = 5 km

2500 g = 2.5 kg

800 ms = 0.8 s

This provides a quick reasonableness check.

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Scientific Notation Makes Conversions Easier

Metric prefixes work particularly well with scientific notation.

Convert:

35 µm → m

Since:

µ = 10⁻⁶

then:

35 µm = 35 × 10⁻⁶ m

Rewrite:

3.5 × 10⁻⁵ m

Similarly:

420 nm = 420 × 10⁻⁹ m

= 4.20 × 10⁻⁷ m

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Multi-Step Example

Convert:

90 km/h → cm/s

First convert kilometres to metres:

90 km/h × (1000 m / 1 km)

Then metres to centimetres:

× (100 cm / 1 m)

Then hours to seconds:

× (1 h / 3600 s)

Calculate:

90 × 1000 × 100 / 3600

= 2500

Therefore:

90 km/h = 2500 cm/s

This demonstrates the power of the conversion-factor method: unwanted units cancel one at a time.

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Worked Physics Example: Kinetic Energy

A moving object has:

mass = 500 g

speed = 36 km/h

Calculate its kinetic energy.

First convert mass:

500 g = 0.500 kg

Convert speed:

36 km/h = 10 m/s

Use:

Eₖ = ½mv²

Substitute:

Eₖ = ½(0.500)(10)²

Eₖ = 25 J

Without converting the units first, the calculation would produce an incorrect result.

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Worked Physics Example: Pressure

A force of:

600 N

acts over an area of:

20 cm²

Find the pressure in pascals.

Use:

p = F/A

But area must be converted to m².

Since:

1 cm² = 10⁻⁴ m²

then:

20 cm² = 20 × 10⁻⁴ m²

= 2.0 × 10⁻³ m²

Now:

p = 600 / (2.0 × 10⁻³)

p = 3.0 × 10⁵ Pa

Notice that treating 20 cm² as 0.20 m² would produce a major error.

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A Reliable Conversion Strategy

For difficult conversions, use this process:

Identify the starting unit

↓

Identify the required unit

↓

Write a known relationship between the units

↓

Create a conversion factor

↓

Arrange it so the unwanted unit cancels

↓

Repeat if more conversions are needed

↓

Calculate

↓

Check the final unit

↓

Check whether the value is reasonable

This method works much better than trying to memorize many separate rules.

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Common Mistakes

Multiplying When You Should Divide

Instead of memorizing "multiply" or "divide," use conversion factors and allow the units to cancel.

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Forgetting to Convert Before Using an Equation

If an equation expects metres and seconds, do not substitute kilometres and hours without considering the units.

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Forgetting That 1 Hour = 3600 Seconds

Remember:

1 h = 60 min

and:

1 min = 60 s

Therefore:

1 h = 60 × 60 = 3600 s

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Treating Area Like Length

Incorrect:

1 m² = 100 cm²

Correct:

1 m² = 10,000 cm²

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Treating Volume Like Length

Incorrect:

1 m³ = 100 cm³

Correct:

1 m³ = 1,000,000 cm³

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Confusing Mass Units

Remember:

1 kg = 1000 g

not:

1 kg = 100 g

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Confusing m and M

milli (m) = 10⁻³

mega (M) = 10⁶

Capitalization matters.

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Forgetting the Final Unit

A numerical answer without its unit may be meaningless.

Instead of:

v = 12

write:

v = 12 m/s

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Blindly Trusting the Calculator

Always ask whether the converted value makes sense.

If:

5 km = 0.005 m

something is clearly wrong.

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Check Your Understanding

1. Convert 4.5 km to metres.

2. Convert 7200 m to kilometres.

3. Convert 85 cm to metres.

4. Convert 32 mm to metres.

5. Convert 4.2 µm to metres.

6. Convert 650 nm to metres.

7. Convert 2.8 kg to grams.

8. Convert 450 g to kilograms.

9. Convert 750 mg to grams.

10. Convert 2.5 h to seconds.

11. Convert 240 s to minutes.

12. Convert 350 ms to seconds.

13. Convert 72 km/h to m/s.

14. Convert 25 m/s to km/h.

15. Convert 3.0 m² to cm².

16. Convert 500 cm² to m².

17. Convert 2.5 L to m³.

18. Convert 400 cm³ to m³.

19. Convert 1.8 g/cm³ to kg/m³.

20. Explain why 1 cm³ is equal to 10⁻⁶ m³ rather than 10⁻² m³.

21. A cyclist travels 12 km in 30 min. Calculate the average speed in m/s.

22. A 600 g object accelerates at 5.0 m/s². Calculate the resultant force.

23. A car travels at 108 km/h. Convert its speed to m/s.

24. A force acts over an area of 50 cm². Explain why the area cannot simply be divided by 100 when converting to m².

25. Describe two ways you could check whether a unit conversion is reasonable.

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Key Terms

• SI units – internationally agreed standard system of measurement units.
• Unit conversion – changing a measurement from one unit to another without changing the physical quantity.
• Conversion factor – ratio of equivalent quantities used to convert units.
• Metric prefix – prefix representing a power-of-ten multiplier.
• kilo (k) – 10³.
• centi (c) – 10⁻².
• milli (m) – 10⁻³.
• micro (µ) – 10⁻⁶.
• nano – 10⁻⁹.
• mega (M) – 10⁶.
• giga (G) – 10⁹.
• Compound unit – unit made from two or more units, such as m/s or kg/m³.
• Dimensional checking – using units to determine whether an equation or calculation is dimensionally consistent.
• Derived unit – unit formed from combinations of SI base units.

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Key Takeaways

• Unit conversions change the way a quantity is expressed, not the quantity itself.
• SI units provide a common measurement system for science.
• Metric prefixes represent powers of ten.
• kilo = 10³
• centi = 10⁻²
• milli = 10⁻³
• micro = 10⁻⁶
• nano = 10⁻⁹
• Conversion factors are a reliable method for both simple and multi-step conversions.
• Arrange conversion factors so unwanted units cancel.
• 1 h = 3600 s
• 1 m/s = 3.6 km/h
• Area conversion factors must be squared.
• Volume conversion factors must be cubed.
• 1 cm³ = 1 mL
• 1 L = 10⁻³ m³
• 1 cm³ = 10⁻⁶ m³
• 1 g/cm³ = 1000 kg/m³
• Physics calculations often require quantities to be converted to compatible SI units before substitution.
• Units can be used to check calculations.
• Always check whether the direction and size of a converted value are reasonable.

A particularly useful habit is:

Don't just convert the number — follow the units.

If the unwanted units cancel and the final value is physically reasonable, you have a strong check that the conversion was performed correctly.