In this problem we are going to explore the evidence for the expansion of the universe that was first discovered by Edwin Hubble. In the $1920$s$,$ Hubble discovered that far$-$off galaxies were all redshifted by an amount that increased with their distance from us$.$ In other words, it appeared that the further away a galaxy was, the faster it was receding from us$.$ This became known as Hubble's law.

Suppose that you attempt to measure the hydrogen spectral lines from three distant galaxies. You get the following values for what corresponds to the $656\mathrm{.}46-$nm hydrogen line. For galaxy $1$ you get a line at $656\mathrm{.}50$ nm $,$ for galaxy $2$ the line is at $656\mathrm{.}57$ nm $,$ and for galaxy $3$ the line is at $656\mathrm{.}64$ nm $.$ Using a few independent techniques $($which we will not go into here$)$ you are able to pin down the distances to these three galaxies. Galaxy $1$ is determined to be approximately ${10}^6$ light years away $($or $1$ Mly $),$ galaxy $2$ is $2\mathrm{.}5$ Mly away, and galaxy $3$ is $4$ Mly away. Recall that a light year is the distance that light travels in a year or roughly ${9\mathrm{.}46}^2{10}^{15}m.$

For this problem we can use the standard special relativistic Doppler shift to determine the galaxies' recessional velocities. One should note that this is not strictly correct because, unlike velocities of objects passing through space, Hubble's law is due to the expansion of space itself, which requires the more complicated theory of general relativity. For cases in which the galaxies are relatively close to earth $($such as this problem$)$ and thus have recessional speeds $v,$ the Doppler shift equations from general relativity become the same $as$ those from special relativity.