Answer
$v = 0.8~c$
Work Step by Step
We can use the expression for relativistic momentum to find the particle's speed:
$p = \gamma ~m ~v$
$p = \frac{m ~v}{\sqrt{1-\frac{v^2}{c^2}}}$
$p~\sqrt{1-\frac{v^2}{c^2}} = m~v$
$p^2~(1-\frac{v^2}{c^2}) = m^2~v^2$
$p^2 = m^2~v^2+\frac{p^2~v^2}{c^2}$
$p^2 = (m^2+\frac{p^2}{c^2})~v^2$
$v^2 = \frac{p^2}{m^2+\frac{p^2}{c^2}}$
$v^2 = \frac{(4.0\times 10^5~kg~m/s)^2}{(1.0\times 10^{-3}~kg)^2+\frac{(4.0\times 10^5~kg~m/s)^2}{(3.0\times 10^8~m/s)^2}}$
$v^2 = \frac{(4.0\times 10^5~kg~m/s)^2}{2.78\times 10^{-6}~kg^2}$
$v = \frac{4.0\times 10^5~kg~m/s}{\sqrt{2.78\times 10^{-6}~kg^2}}$
$v = 2.4\times 10^8~m/s$
$v = 0.8~c$