State the principle of homogeneity. Test the dimensional homogeneity of equations -
(i) `s = ut + 1/2 at^(2)` (ii) `S_(n) = u + a/2 (2n - 1)`

Explanation

(i) Dimension of L.H.S
[s] = `[M^(0)L^(1)T^(0)]`
Dimension of R.H.S. = `[ut] + [at^(2)]`
= `[LT^(-1).T] + [M^(0)L^(1)T^(-2).T] = [M^(0)L^(1)T^(0)]`
as Dimension of L.H.S = Dimension of R.H.S
`:.` The equation is dimensionally homogeneous.
(ii) `S_(n) =` Distance travelled in `n^(th)` sec that is `(S_(n) - S_(n-1))`
`:. S_(n) = u xx 1 + a/2 [2n - 1]`
`[LT^(-1)] = [LT^(-1)] + [LT^(-2)][T]`
`[LT^(-1)] = [LT^(-2)]`
L.H.S. = R.H.S.
Hence this is dimensionally correct.