Answer
The galaxy is receding from the earth with a speed of $1.5\times 10^6~m/s$
Work Step by Step
Since the wavelength of the light from the galaxy is longer than the wavelength measured in a laboratory, the galaxy is receding from the earth.
We can find the speed of the galaxy as:
$\lambda' = \lambda_0~\frac{\sqrt{1+v/c}}{\sqrt{1-v/c}}$
$1.005~\lambda_0 = \lambda_0~\frac{\sqrt{1+v/c}}{\sqrt{1-v/c}}$
$1.005 = \frac{\sqrt{1+v/c}}{\sqrt{1-v/c}}$
$(1.005)^2 = \frac{1+v/c}{1-v/c}$
$(1.005)^2(c-v) = c+v$
$(1.005)^2~v + v = (1.005)^2~c - c$
$v = \frac{(1.005)^2 - 1}{(1.005)^2 + 1}~c$
$v = \frac{(1.005)^2 - 1}{(1.005)^2 + 1}~(3.0\times 10^8~m/s)$
$v = 1.5\times 10^6~m/s$
The speed of the galaxy is thus $1.5\times 10^6~m/s$.
