By method of dimensional analysis, derive the relation:

S=ut+1/2at², where the letters have their usual meanings

Limitation of the dimensional analysis is that we can not derive the exact formula. So we can not derive S = ut +1/2at²
But we can check the correctness of the formula.

S = ut + 1/2 at²
LHS:

S = distance
unit = meter
Dimension = $\left[M^0{L}^1{T}^0\right]$

RHS:
u = initial velocity
 unit = m/s
Dimension = $\left[M^0{L}^1{T}^{-1}\right]$

t = time
unit = second
Dimension = $\left[M^0{L}^0{T}^1\right]$

Dimension of ut = $\left[M^0{L}^1{T}^{-1}\right]$ $\left[M^0{L}^0{T}^1\right]$ = $\left[M^0{L}^1{T}^0\right]$
This is the dimension the distance.

a = acceleration
unit = m/s²
Dimension = $\left[M^0{L}^1{T}^{-2}\right]$

Dimension of at² = $\left[M^0{L}^1{T}^{-2}\right]$ $\left[M^0{L}^0{T}^2\right]$ = $\left[M^0{L}^1{T}^0\right]$
This is the dimension of distance.

Hence dimensions of the LHS and RHS are equal.