Answer
$v = 0.0014~c$
$v = 4.2\times 10^5~m/s$
Work Step by Step
We can find $v$ when $\gamma = 1.000001$
$\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} = 1.000001$
$1-\frac{v^2}{c^2} = \frac{1}{(1.000001)^2}$
$\frac{v^2}{c^2} = 1-\frac{1}{(1.000001)^2}$
$v^2 = [1-\frac{1}{(1.000001)^2}~]~c^2$
$v = \sqrt{1-\frac{1}{(1.000001)^2}}~\times c$
$v = 0.0014~c$
$v = (0.0014)~(3.0\times 10^8~m/s)$
$v = 4.2\times 10^5~m/s$
