Question

using dimensional method show that the velocity v of a body falling freely under gravity depends on _/gh where g is the acceleration due to gravity and h is the height of the body

Answer 1

Let $v=g^a{h}^b$ ..........(i)
Where a and b are the powers of g and h respectively to represent v.
Writing the dimensions in terms of M, L, T on either side of equation (i), we get
$$\begin{gathered} \left[M^0{L}^1{T}^{-1}\right]={\left(M^0{L}^1{T}^{-2}\right)}^a{L}^b \\ =M^0{L}^a{T}^{-2a}{L}^b \\ =M^0{L}^{a+b}{T}^{-2a} \end{gathered}$$

Applying the principle of homogeneity of dimensions, we get

$$\begin{gathered} 0=0 \\ a+b=1..........\left(ii\right) \\ -2a=-1ora=\frac{1}{2} \end{gathered}$$
Putting the value of a in equation (ii), we get

$$\begin{gathered} \frac{1}{2}+b=1 \\ or,b=\frac{1}{2} \end{gathered}$$
Now putting the value of a and b in equation (i), we get

$$\begin{gathered} v=g^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{h}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} \\ or,v=\sqrt{gh} \\ or,v^2=gh \end{gathered}$$