## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th Edition)

$v = 0.866~c$
We can write an expression for the rest energy: $E = mc^2$ We can write an expression for the total energy: $E = \gamma ~mc^2$ If the total energy is twice the rest energy, then $\gamma = 2$ We can find the speed when $\gamma = 2$: $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ $\sqrt{1-\frac{v^2}{c^2}} = \frac{1}{\gamma}$ $1-\frac{v^2}{c^2} = \frac{1}{\gamma^2}$ $\frac{v^2}{c^2} = 1-\frac{1}{\gamma^2}$ $v^2 = (1-\frac{1}{\gamma^2})~c^2$ $v = \sqrt{1-\frac{1}{\gamma^2}}~c$ $v = \sqrt{1-\frac{1}{2^2}}~c$ $v = \sqrt{0.75}~c$ $v = 0.866~c$