Question:
Theory of relativity reveals that mass can be converted into energy. The energy $E$ so obtained is proportional to certain powers of mass $m$ and the speed of light $c$. Guess a relation among the quantities using the method of dimensions.
Solution:
Let us assume energy $E \alpha m^{a} c^{b}$, or $E=k m^{a} c^{b}$, where $k$ is a constant Equating their dimensions,
$\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]=\left[\mathrm{M}^{\mathrm{a}}\right]\left[\mathrm{L}^{\mathrm{b}} \mathrm{T}^{-\mathrm{b}}\right]$
From here, we see that $a=1$ and $b=2$