prove with the help of dimensional analysis that the equation h= 1/2 gt for a distance travelled by a body falling freely in time t is incorrect. Find the correct equation with the help of dimension.

[L] = [L T^-2][T]
[L] = [L T^​-1]
SO,THE EQUATION IS DIMENSIONALLY INCORRECT .
h = 1/2 g^a t^b
1/2 gets removed so the equation=
h = g^at^b
[L T⁰] =[L T^-2]^a [T]^b 
[L T^​0] =[L^a T^{-2 a}​] [T^b]
[LT⁰] = [L^a T^{-2 a + b} ]
a=1                                                               (1)
-2a+b = 0                                                 (2)
installing the value of equation 1 in equation 2 , we get,
-2*1+b = 0
b=2
so,
a=1 , b=2
h = 1/2 g^at^b
h = 1/2 g¹t²
now, the final equation is=
h=1/2 gt² .