Question

At what speed, as a fraction of $c$, is a particle’s kinetic energy twice its rest energy?

Short answer

$v=0.88c$

Step-by-step solution

Given Information

We have to find the speed $v$, as a fraction of $c$, is a particle’s kinetic energy twice its rest energy.

Simplify

Total energy formula is

$E=m{c}^2\Rightarrow E=\frac{m{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}$

And

The rest of energy formula is

$E_o=m{c}^2$

To find speed $v$, as a fraction of $c$,we will calculate it as below

$$\begin{gathered} E=2E_o \\ \frac{m{c}^2}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}=2m{c}^2 \\ \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=2 \\ \frac{v}{c}=\sqrt{1-{\left(\frac{1}{2}\right)}^2} \\ \frac{v}{c}=0.88 \\ v=0.88c \end{gathered}$$