# Do I use Relativistic Energy here?

1. Oct 13, 2008

### mattst88

1. The problem statement, all variables and given/known data

What is the speed of a proton when its kinetic energy is equal to its rest energy?

2. Relevant equations

$$K = mc^2(\gamma - 1)$$
$$E_0 = mc^2$$
$$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

3. The attempt at a solution

$$K = E_0$$
$$mc^2(\gamma - 1) = mc^2$$
$$\gamma = 2$$
$$\frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = 2$$
$$0.5 = \sqrt{1 - \frac{v^2}{c^2}}$$
$$0.5^2 = 1 - \frac{v^2}{c^2}$$
$$c^2 \sqrt{1 - 0.25} = v^2$$
$$v = c \sqrt{0.75} = 0.866 c$$