Dimensional analysis


 * See also: Natural units


 * From Dimensional analysis:

Any physical law that accurately describes the real world must be independent of the units (e.g. km or mm) used to measure the physical variables.

Consequently, every possible commensurate equation for the physics of the system can be written in the form


 * $$a_0 \cdot D_0 = (a_1 \cdot D_1)^{p_1} (a_2 \cdot D_2)^{p_2}...(a_n \cdot D_n)^{p_n}$$

The dimension, Dn, of a physical quantity can be expressed as a product of the basic physical dimensions length (L), mass (M), time (T), electric current (I), absolute temperature (Θ), amount of substance (N) and luminous intensity (J), each raised to a rational power.

Suppose we wish to calculate the when fired with a vertical velocity component $$V_\mathrm{y}$$ and a horizontal velocity component $$V_\mathrm{x}$$, assuming it is fired on a flat surface.

The quantities of interest and their dimensions are then


 * $range$ as L$x$


 * $$V_\mathrm{x}$$ as L$x$/T


 * $$V_\mathrm{y}$$ as L$y$/T


 * $g$ as L$y$/T2

The equation for the range may be written:


 * $$range = (V_x)^a (V_y)^b (g)^c$$

Therefore


 * $$\mathsf{L}_\mathrm{x} = (\mathsf{L}_\mathrm{x}/\mathsf{T})^a\,(\mathsf{L}_\mathrm{y}/\mathsf{T})^b (\mathsf{L}_\mathrm{y}/\mathsf{T}^2)^c\,$$

and we may solve completely as $$a=1$$, $$b=1$$ and $$c=-1$$.