The rest energy of a particle is
[tex]E_0=m_0 c^2 [/tex]
where [tex]m_0[/tex] is the rest mass of the particle and c is the speed of light.
The total energy of a relativistic particle is
[tex]E=mc^2 = \frac{m_0 c^2}{ \sqrt{1- \frac{v^2}{c^2} } } [/tex]
where v is the speed of the particle.
We want the total energy of the particle to be twice its rest energy, so that
[tex]E=2E_0[/tex]
which means:
[tex] \frac{m_0c^2}{ \sqrt{1- \frac{v^2}{c^2} } }=2m_0 c^2 [/tex]
[tex] \frac{1}{ \sqrt{1- \frac{v^2}{c^2} } }=2 [/tex]
From which we find the ratio between the speed of the particle v and the speed of light c:
[tex] \frac{v}{c}= \sqrt{1- (\frac{1}{2})^2 } =0.87 [/tex]
So, the particle should travel at 0.87c in order to have its total energy equal to twice its rest energy.
