Question

The relativistic mass of a proton moving with speed of 2.4×10~8 m/s (rest mass of proton=1.61×10~-27kg and c= 3×10~8m/s) is?

Answer 1

The relativistic mass of a body of rest mass $m_0$ is

$m=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$, where $v$ is the speed of the body and $c$ is the speed of light.

For a proton, $m_0=1.61*10^{-27}\; kg, c=3*10^8\;m/s,v=2.4*10^8\;m/s$.

Substituting numerical values,

$m=\frac{1.61*10^{-27}}{\sqrt{1-\frac{(2.4*10^8)^2}{(3*10^8)^2}}} \\ m=2.683*10^{-27}\;kg$

Answer 2

As per Einstein's theory of relativity, the mass of a body increases when its speed is comparable to the velocity of light.

Let us consider the rest mass of a body is $m_{0}$

The mass of the body when it will move with a velocity v is -

                                           $m\ =\frac{m_{0}} {\sqrt{1-\frac{v^2}{c^2} }}$

Here, c is the velocity of light and its value is  $3*10^8\ m/s$

As per the question, we have proton.

The rest mass of proton is  $1.61*10^-27\ kg$

The velocity of proton  $v=\ 2.4*10^{8}\ m/s$

Hence, the relativistic mass of a proton -

                                         $m\ =\frac{1.61*10^-27\ kg}{\sqrt{1-\frac{[2.4*10^8]^2}{[3*10^8]^2}}}$

                                                $=\frac{1.61*10^-27\ kg}{\sqrt{1-0.64}}$

                                                $=\frac{1.61*10^-27\ kg}{\sqrt{0.36}}$

                                                $=\frac{1.61*10^-27\ kg}{0.6}$

                                                $= 2.68*10^-27\ kg$   [ans]