## Dimensional Analysis of the Kinematic Equations

To help to remember the kinematic equations, it may be useful to know a little dimensional analysis.

The kinematic equations include:

[1] or [¬s]: $v=u+at$

[2] or [¬v]: $s = ut + \frac{1}{2}at^2$

[3] or [¬u]: $s = vt - \frac{1}{2}at^2$

[4] or [¬a]: $s=\dfrac{(u+v)t}{2}$

[5] or [¬t]: $v^2=u^2+2as$

where

$s$ = position (or distance)
$u$ = initial velocity
$v$ = final velocity
$a$ = acceleration
$t$ = time (or duration)

As I’ve mentioned in an earlier post: like the complexity notation in computing, ‘dimension formulae’ are just notation, and not really formulae; likewise, (numerical) constant factors are ignored.

The dimensions of interest here are:
$L$ = length (space);
$T$ = time.

Taking the formulae in turn:

[1]: $LT^{-1} = LT^{-1}+LT^{-2}\times T$,
so the terms are all velocities, $LT^{-1}$;

[2]: $L = LT^{-1}\times T + LT^{-2}\times (T)^2$,
so the terms are all lengths, $L$;

[3]: $L = LT^{-1}\times T - LT^{-2}\times (T)^2$;

[4]: $L = (LT^{-1} + LT^{-1})\times T$,
so the terms are all $L$;

[5]: $(LT^{-1})^2 = (LT^{-1})^2+LT^{-2} \times L$,
so the terms are all $L^2T^{-2}$;

It can be seen that these formulae are dimensionally correct. Only like terms are added or subtracted; both sides of the equation are of like dimension.

Note that when using this for other formulae in physics, the physical constants may have dimensions. (For example, the speed of light, although constant, does have the dimensions of velocity.) Also note that even a zero may have dimensions (no metres is different from no seconds).

Dimensional analysis will not show that a formula is correct, but may show if it is wrong: if it is dimensionally incorrect, then it is just incorrect.

One thing that dimensional analysis won’t do is determine numerical constants. However, if you’re at all familiar with calculus, you might recall that

$\int t \, dt = \frac12 t^2$

Thus, if you see the $t^2$ in formula [2], you might remember that it came from an integration, and so remember the $\frac12$.

(See also my earlier posts on dimensional analysis of statistical formulae and kinematic equations.)