Question

At what fraction of the speed of light does a particle travel if it's kinetic energy is twice it's rest mass

Answer 1

The total energy of the particle is $E_{tot} = E_{kin}+E_{rest}=2E_{rest}+E_{rest}=3E_{rest}$.

At the same time the special relativity gives us the relations $E_{tot.} = \frac{mc^2}{\sqrt{1-v^2/c^2}}$, $E_{rest}=mc^2$.

From this we deduce $\sqrt{1-v^2/c^2} = 1/3, v^2/c^2= 8/9$ and thus $\frac{v}{c}=\frac{2\sqrt{2}}{3}$.