Correct Answer - Option 4 : √3 : 2
Concept:
Variation of Mass with Velocity:
According to classical physics, the inertial mass of a body is independent of the velocity of light. It is regarding as a constant. However special theory of relativity leads us to the concept of variation of mass with velocity. It follows from the special theory of relativity that the mass m of a body moving with relativistic velocity v relative to an observer is larger than its m0 when it is at rest.
According to Einstein, the mass of the body in motion is different from the mass of the body at rest.
\(m = \frac{{{m_o}}}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}\) ---(1)
This is the relative formula for variation of mass with velocity
m0 is the rest mass
m is the relativistic mass of the body
v = relativistic velocity
The total energy of a moving particle is given by:
\(E=mc^2 = \frac{{{m_oc^2}}}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}\)
where m0 c2 = Rest mass-energy
Calculation:
Given E = 2 × m0c2
\(2 × m_0c^2= \frac{{{m_oc^2}}}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}\)
Rearranging the above, we get:
\(1 - \frac{{{v^2}}}{{{c^2}}}=\frac{1}{4}\)
\(\frac{{{v^2}}}{{{c^2}}}=\frac{3}{4}\)
\(\frac{{{v}}}{{{c}}}=\frac{\sqrt 3}{2}\)
