Question 6.1: The vertical displacement h of a freely falling body from it...

The above phenomenon can be described by the functional relation as

$F(h, t, g)=0$                   (6.7)

Here , the number of variables m = 3 (h, t, and g) and they can be expressed in terms of two fundamental dimensions L and T. Hence, the number ofπterms = m – n = 3 – 2 = 1. In determining thisπterm, the number of repeating variables to be taken is 2. Since h is the dependent variable, the only choice left for the repeating variables is with t and g.

Therefore,

$\pi_{1}=t^{a} g^{b} h$                (6.8)

By substituting the fundamental dimensions of the variables on the left- and- righthand sides of Eq. (6.8), we get

Equating the exponents of T and L on both the sides of the above equation we have

And $b+1=0$

which give,

Hence, $\pi_{1}=h / g t^{2}$

Therefore the functional relationship (Eq. (6.7)) of the variables describing the phenomenon of free fall of a body under gravity can be written in terms of the dimensionless parameter $\left(\pi_{1}\right)$  as

$f\left(\frac{h}{g t^{2}}\right)=0$            (6.9)

From elementary classical mechanics we know that $\frac{h}{g t^{2}}=\frac{1}{2}$. One should know, in this context, that the Pi theorem can only determine the pertinent dimensionless groups
describing the problem but not the exact functional relationship between them.