Checking equations are homogeneous

4 minAS Level1.2.21.2.3

AI-drafted, independently checked · not yet teacher-reviewed

By the end of this lesson

By the end you can use SI base units to check whether an equation is homogeneous, find the base units of an unknown constant, and explain why a homogeneous equation can still be wrong.

Definition

Homogeneous equation

An equation in which every term, on both sides, has the same SI base units.

What counts as a "term"?

Terms are the chunks that are added, subtracted or set equal. In

$s = ut + \tfrac{1}{2}at^2$

there are three terms: $s$ , $ut$ and $\tfrac{1}{2}at^2$ . Each one must come out in metres.

Inside a term, quantities multiply, so their units multiply too: $ut$ has units $\text{m s}^{-1} \times \text{s} = \text{m}$ .

You can't add apples to seconds

$3\ \text{m} + 2\ \text{s}$ is meaningless, like adding 3 apples to 2 bus timetables. Nature doesn't do that, so a correct physics equation can never contain it. Homogeneity checking is just hunting for apples-plus-timetables.

Worked example: Checking v² = u² + 2as

Show that $v^2 = u^2 + 2as$ is homogeneous.

Have a go first?

Quick check

Multiple choiceQuestion 1 · Apply

Which equation is not homogeneous? ( $P$ is power, $F$ force, $v$ speed, $p$ pressure, $\rho$ density, $g$ acceleration of free fall, $h$ height, $E$ energy, $m$ mass, $a$ acceleration, $s$ displacement.)

P = Fv

p = \rho g h

E = \tfrac{1}{2}mv

v^2 = 2as

Answer the question above to continue.

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