Physics

Derive an expression of kinetic energy of a body of mass 'm' and moving  with velocity 'v' , using dimensional analysis.


SOLUTION

Since we  know that,

The dimension of energy $$[E]=[M^1L^2T^{-2}]$$
The dimension of mass   $$[M]=[M^1L^0T^0]$$
Dimension of velocity$$[V]=[M^0L^1T^{-1}]$$

Let $$[E]=k[M]^x[V]^y$$

Where k is the proportionality constant which is a dimensionless quantity.
Therefore,
$$[M^1L^2T^{-2}]=k[M^1L^0T^0]^x[M^0L^1T^{-1}]^y$$

So, $$x=1,y=2$$

Thus, We have
$$E=kmv^2$$

It is found that $$k=\dfrac{1}{2}$$

Hence, $$E=\dfrac{1}{2}mv^2$$
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