# Dimensional Equations

Dimensions play an important role -> equation validity!

It doesn't make sense to say that: 1 meter = 25 seconds!

Conversely, we already said that 3.2808 feet = 1 meter is valid.
Why?
[hint]

### Dimensional Homogeneity

The dimensions on both sides of the equals sign must be the same
for an equation to be valid. Another way to say this is that valid
equations must be dimensionally homogeneous.

It is a good practice to make **units** the same (via
conversion).

It is also good practice (and a course requirement!) to show all
units throughout a problem to test equation validity!

OUTCOME:

Identify an invalid equation, based on
dimensional arguments

TEST YOURSELF!

Which of these equations is dimensionally
homogeneous?

$x(m) = x_o(m) + 0.3048(m/ft)v(ft/s)t(s) +
\frac{1}{2}a(m/s^2)[t(s)]^2$

$P\left (\frac{kg}{ms^2} \right ) = 101325.0 \left
(\frac{Pa}{atm} \right )1\left (\frac{\frac{kg}{ms^2}}{Pa}
\right ) P_o (atm) + \rho (kg/m^3)v(m/s)$

NOTE:

It is sometimes simpler to write the units as
"vertical fractions" to facilitate canceling.

IMPORTANT:

Just because an equation is dimensionally
homogeneous does **not** mean that it is valid! Dimensional
considerations act as a first test for validity only!