A student recalls the relativistic mass formula almost correctly but forgets where to put the speed

Question

A student recalls the relativistic mass formula almost correctly but forgets where to put the speed of light $c$. He writes $m = \frac{m_0}{(1 - v^2)^{1/2}}$. Where should the missing $c$ be placed to make the equation dimensionally correct?

Professor PC Thomas Classes · Answer

The term inside the bracket $(1 - v^2)$ must be dimensionless because $1$ is a dimensionless number. Thus, $v^2$ must be divided by $c^2$ to make the whole term dimensionless. The correct relation is $m = \frac{m_0}{(1 - v^2/c^2)^{1/2}}$.