Metre

The metre ($\mathrm{m}$) is the SI base unit for length.

It is defined by fixing the speed of light in vacuum as exactly:

$$c=299792458\mathrm{m}{\mathrm{s}}^{-1}$$

Because the Second is already defined, this fixes the metre. One metre is the length of the path travelled by light in vacuum during:

$$\frac{1}{299792458}\mathrm{s}$$

So the definition is not “a metal bar in Paris”; it is a relationship between time and the invariant speed of light.

Useful intuition:

• Light travels about $3.00\times 10^8\mathrm{m}$ in one second.
• Light travels about $0.30\mathrm{m}$ in one nanosecond.
• A metre is therefore roughly the distance light travels in $3.34\mathrm{ns}$.

Dimensional role: length is usually written $L$, and the metre appears in Velocity as $\mathrm{m}{\mathrm{s}}^{-1}$, Newton as $\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2}$, and Joule as $\mathrm{kg}{\mathrm{m}}^2{\mathrm{s}}^{-2}$.

Common mistake: treating $c=299792458$ as just a number. It carries units; the fixed value defines the scale of metres relative to seconds.

Quick check: centimetres and kilometres are convenient prefixes, but dimensional analysis should reduce them back to metres before combining equations.

Quick check

Metres measure length, but powers of metres encode geometry:

• $\mathrm{m}$: length
• ${\mathrm{m}}^2$: area
• ${\mathrm{m}}^3$: volume

This is why dimensional analysis catches missing factors of radius, area, or volume in mechanics and electromagnetism problems.