Question

A famous relation in physics relates ‘moving mass’ $m$ to the ‘rest 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 special relativity due to Albert Einstein). A boy recalls the relation almost correctly but forgets where to put the constant $c$. He writes:
$m = \dfrac{{{m_0}}}{{{{\left( {1 - {v^2}} \right)}^{1/2}}}}$
Guess where to put the missing $c$.

Answer 1

Nothing in the universe can match the speed of light. The velocity of the particle as mentioned in the question has velocity much smaller than the speed of light. We know that square root, a negative number, can't be shown in real form. So, to remove this anomaly, we will try to make the value under the square root, less than one.

Formula used:
Using relativistic mass formula,
${m_{rel}} = \dfrac{{{m_0}}}{{\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} }}$ …… (1)
Where,
${m_0}$ is mass
$v$ is speed
$c$ is the total quantity of energy.

Complete answer:
Amounts inside functions (square root here should be dimensionless). The $\left[ {1 - {v^2}} \right]$ definition should be positive and the quantity will be dimensionless. Square root of a negative number does not exist in real form. As we know that the velocity of the particle is much smaller than the speed of the light. So, the value of the fraction $\dfrac{{{v^2}}}{{{c^2}}}$ is obviously less than one. To keep the quantity in real form, we will put the constant $c$ as a denominator.

Thus, the correct formula is ${m_{rel}} = \dfrac{{{m_0}}}{{\sqrt {1 - \dfrac{{{v^2}}}{{{c^2}}}} }}$.

Additional information:
Relativistic mass: Relativistic mass, the mass that is attributed to a body in motion in the special theory of relativity. If the velocity of the body reaches the speed of light, the relativistic mass m becomes infinite, because even though great momentum and energy are supplied arbitrarily to a body, its velocity still stays lower than $c$.

Note: Remember that to describe electron orbital contraction in heavy elements, relativistic mass is used. The principle of mass as a property of a Newtonian mechanical object does not have a direct connection to the concept of relativity.