|MadSci Network: Astronomy|
There is no exact answer to your question, but I can nevertheless provide some information.
The rate of expansion of the universe is specified by a number called the Hubble constant. It has a value of about 72 kilometers per second per million parsecs of distance. In more familiar units, this is 22 kilometers per second per million light years of distance. One parsec is about 3.26 light years.
This means that two points in space separated by (say) 10 million light years will be moving apart at a relative speed of 220 kilometers per second. Two points separated by 20 million light years will move apart at a speed of 440 kilometers per second. The rate of separation increases with distance, as Edwin Hubble discovered quite some time ago.
Not everything expands as the universe expands. Any gravitationally bound object will not expand and will be held together by its own gravity. Here is a list of things that are not themselves expanding: the Earth, planets, the Sun and other stars, our Galaxy and other galaxies, the Virgo cluster and other galaxy clusters.
Now things that are widely separated will be moving apart because of the expansion of the universe. Consider the Virgo supercluster (a cluster of galaxies) and some other, very distant cluster of galaxies. Neither will be expanding, but both will be moving apart as the universe expands.
One way to think of this is to imagine that galaxy clusters are scattered though space like a large number of coins on a tile floor. If the tiles were getting bigger, the coins would (on average) be moving apart. The greater the separation between two coins, the faster they will move apart.
The complication which makes the answer less than simple is that galaxies are moving around through space in more or less a random way. This is in addition to the relative motion that they have from the expansion of the universe. So perhaps it would be better to think of little insects on a tile floor. If the tiles are expanding and the bugs are (slowly) moving around, then you would have a feel for what galaxy motions are like in an expanding universe.
For nearby galaxies, the random motion is far larger than the motion provided by the expansion of the universe. For example, our galaxy and the Andromeda galaxy are approaching one another. If you look far off, however, the expansion of the universe gives galaxies a much larger speed than the random motion.
How far is this? The random motion is about 400 kilometers per second. In order for the expansion to be a much bigger effect, we have to look at galaxies that are far enough away that the expansion imparts a relative velocity of more than 400 kilometers per second. A distance of 30 million parsecs will do adequately: at that distance, galaxies are moving away from us (on average) with a speed of over 2,000 kilometers per second.
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