A famous relation in physics relates the moving mass $m_{0}$ of a particle in terms of its speed v and the speed of light c. (This relation first arose as a consequence of the special theory of relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant c. He writes $m = \frac{m_{0}}{{(1 - v^{2})}^{1 / 2}}$ . Guess where to put the missing c.

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Solution

According to the principle of homogeneity of dimensions, powers of M, L, T on either side of the formula must be equal. For this, on RHS, the denominator $(1 - v^{2})^{1 / 2}$ should be dimensionless. Therefore, instead of $(1 - v^{2})^{1 / 2}$ ,we should write $(1 - v^{2} / c^{2})^{1 / 2}$ . Hence, the correct formula would be
$m = \frac{m_{0}}{(1 - v^{2} / c^{2})^{1 / 2}}$

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