Answer
$v = 2.8\times 10^8~m/s$
Work Step by Step
We can find the electron's speed:
$p = \gamma~m~v$
$v = \frac{p}{\gamma~m}$
$v = \frac{p}{\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}~m}$
$v = \frac{p}{m}~\sqrt{1-\frac{v^2}{c^2}}$
$v^2 = \frac{p^2}{m^2}~(1-\frac{v^2}{c^2})$
$1 = \frac{p^2}{v^2~m^2}~-\frac{p^2}{m^2~c^2}$
$\frac{p^2}{v^2~m^2} = 1+\frac{p^2}{m^2~c^2}$
$\frac{p^2}{v^2~m^2} = \frac{m^2~c^2}{m^2~c^2}+\frac{p^2}{m^2~c^2}$
$\frac{v^2~m^2}{p^2} = \frac{m^2~c^2}{m^2~c^2+p^2}$
$v^2 = \frac{p^2~c^2}{m^2~c^2+p^2}$
$v = \frac{p~c}{\sqrt{m^2~c^2+p^2}}$
$v = \frac{(2.4\times 10^{-22}~kg~m/s)(3.0\times 10^8~m/s)}{\sqrt{(9.1\times 10^{-31}~kg)^2~(3.0\times 10^8~m/s)^2+(2.4\times 10^{-22}~kg~m/s)^2}}$
$v = 2.8\times 10^8~m/s$
