Question

Theory of relativity reveals that mass can be converted into energy. The energy E so obtained is proportioanl to certain powers of mass m and the speed c of light. Guess a relation among the quantities using the method of dimensions.

Answer 1

Let energy $E\propto M^a{C}^b$ where $M=$Mass, $C=$ speed of light
$\Rightarrow E=K{M}^a{C}^b(K=$ proportionality constant)
Dimension of left side
$E=\left[M{L}^2{T}^{-2}\right]$
Dimension of right side
$M^a=[M{]}^a,[C{]}^b=[L{T}^{-1}{]}^b$
$\therefore \left[M{L}^2{T}^{-2}\right]=\left[M{]}^a\right[L{T}^{-1}{]}^b$
$\Rightarrow a=1;b=2$
So, the relation is $E=KM{C}^2$.