Answer
$v = 0.866~c$
Work Step by Step
The Newtonian momentum is $~~p = m~v$
The relativistic momentum is $~~p = \gamma ~m~v$
We can find the speed when $\gamma = 2$:
$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
$\sqrt{1-\frac{v^2}{c^2}} = \frac{1}{\gamma}$
$1-\frac{v^2}{c^2} = \frac{1}{\gamma^2}$
$\frac{v^2}{c^2} = 1-\frac{1}{\gamma^2}$
$v^2 = (1-\frac{1}{\gamma^2})~c^2$
$v = \sqrt{1-\frac{1}{\gamma^2}}~c$
$v = \sqrt{1-\frac{1}{2^2}}~c$
$v = \sqrt{0.75}~c$
$v = 0.866~c$
