What happens to the units when squaring a variable?

What happens to the units of a squared variable?

For example, if I squared velocity, would the units, metres per second (${\rm m}/{\rm s}$), change as well?

Yes. If you square a variable, its unit of measurement is also squared, in the case of speed $v$ in $m/s$ ($ms^{-1}$), then $v^2$ is expressed in $m^2s^{-2}$. This is true for all physical variables (or constants).

Yes. Consider the equation for kinetic energy (KE):

$${\rm KE} = \frac{1}{2} mv^{2}$$

the dimensions of KE are:

$${\rm mass} \times {\rm velocity}^{2}=\frac{{\rm mass} \times {\rm length}^{2}}{{\rm time}^{2}}$$

or with SI units:

$$1\,{\rm J} = 1\,{\rm kg}\,{\rm m}^{2}\,{\rm s}^{-2}$$

• Cleaned up your mathjax quite a bit (have a look at 'edit' if you want to see the source). Hope you agree that it looks a bit better. Aug 26, 2015 at 18:22
• @KyleOman Nice I am new at Latex Aug 26, 2015 at 18:24
• The '\rm' means 'roman', as in upright font (as opposed to 'italic'). Typically format units as roman and variables as italic in formulae, and text looks way better in roman as well. Cheers :) Aug 26, 2015 at 18:25

Yes.The unit of $(\text{velocity})^2$ is $[\frac{\text{m}}{\text{s}}]^2$ .This is true for all calculations for any physical quantity.On squaring a physical quantity, its dimension gets squared. As a result, the unit is also squared.

A slight expansion of the question:

When a physical quantity in an equation is raised to a power (like being squared) then all the physical quantities that go into that quantity are also raised to the same power.

So, velocity, $v$, has dimensions of length over time, $l/t$, and velocity squared, $v^2$, has dimensions of length squared over time squared, $l^2/t^2$.

As a consequence of this, the units used for these quantities follow the same rules.

Additionally, in any physics formula the physical quantities on the right hand side must equal the physical quantities on the left hand side.