Check The correctness of the following equation using dimensional analysis. Make a comment on it.
`S = ut + 1//2 "at"^(2)` where s is the displacement, u is the initial velocily, t is the time and a is the acceleration produced,

Explanation

Dimension for distances s = [L]
Dimension or initial velocity u = `[LT^(-1)]`
Dimension for time t = [T]
Dimension for acceleration a = `[LT^(-2)]`
According to the principle of homogeneity,
Dimensions of LHS = Dimensions of RHS
Substituting the dimensions in the given formula
`S = ut + 1//4 at^(2), 1/4` is a number. It has no dimensions
[L] = `[LT^(-1)][T^(1)] + [LT^(-2)][T^(2)]`
[L] = [L] + [L]
As the dimensional formula of LHS is same as that of RHS, the equation is dimensionally correct.
Comment: But actually it is a wrong equation. We know that the equation of motion is `s = ut + 1//2 at^(2)`
So, a dimensionally correct equation need not be the true (or) actual equation But a true equation is always dimensionally correct.