Question

Show by dimensional analysis that this expression is satisfied and can this analysis give the value of k?
The position of a particle moving under uniform acceleration is some function of time and the acceleration. Suppose we write this position
$s=k{a}^m{t}^n$
Where, k is a dimensionless constant. Show by dimensional analysis that this expression is satisfied if m=1 and n=2. Can this analysis give the value of k?

Answer 1

Hint: From the above provided expression we could first identify the physical quantities. Then we could simply perform dimensional analysis for the expression on either side. You could then find the value of m and n by using the principle of homogeneity. Thereby, check whether the value of k could be found.

Complete answer:
In the question, we are given the expression,
$s=k{a}^m{t}^n$
We know the dimensions of these quantities as,
$S\Rightarrow \left[L\right]$
$a\Rightarrow \left[L{T}^{-2}\right]$
$t\Rightarrow \left[T\right]$
k is a dimensionless constant. Dimensional analysis to the above equation,
$\left[L\right]={\left[L{T}^{-2}\right]}^m{\left[T\right]}^n$
${\left[L\right]}^1{\left[T\right]}^0={\left[L{T}^{-2}\right]}^m{\left[T\right]}^n$
By applying the principle of homogeneity we could equate the powers of same quantities to get,
$m=1$
$-2m+n=0$
$\Rightarrow n=2$
Therefore, we proved that the above given expression is satisfied if m=1 and n=2. Also, we clearly gave the question that k is a dimensionless constant and we are well aware of the fact that dimensional analysis will not give the value of dimensionless constant.

Note:
The statement of principle of homogeneity goes like this, ‘the dimensions of each of the terms belonging to a dimensional equation on either side should be the same. The study of the relation between various physical quantities present in an equation based on their respective units and dimensions are called dimensional analysis.