What is the principle of homogeneity …

Principle of Homogeneity of Equation/Dimensions -

An equation  relating physical quantities must have the same dimensions on left hand side as on the right hand side.

For example, if we know displacement, s, depends upon initial velocity, u, acceleration,a and time, t.

and ask ourselves what must be the equation connecting all these. The principle of homogeneity tells us that

if we write, displacement s on left hand side then on the right hand side, u, a and t must be combined to get displacement dimensions or units. Hence

s = ut + at²

[L] = [LT ^-1] [T] + [LT ^-2] [T]²

[L] = [L]  + [L]

So that means we can add a length to another length.

And we'll the resultant quantity with the dimensions of length.

As we can see the principle of homogeneity tells us the form of the equation, the exact equation is obtained by theoretical analysis,

and we know that the correct equation is,

s = ut + at²/2