The Milky Way and Andromeda are plunging towards each other, aimed nearly directly at each other, and proceeding at a pace of about 110 kilometers every second. Given the enormous distance between our two galaxies, these two will still take some three billion years to bridge the space between them, even though 110 kilometers every second would get you from New York to Tokyo in about a minute and a half. At that speed, you could travel to the Moon in about an hour, and jet between the Earth and Pluto in about two years. By comparison, New Horizons’ journey took nearly 10 years to cross that same distance.
This is not a particularly rapid pace for a collision between galaxies - most studies of interactions between two galaxies (at least in the relatively nearby universe) choose galaxies which are moving at less than 300 kilometers per second, relative to their companion. Even then, the typical encounters happen at a relatively slow pace, slightly less than 100 km/s. If you’re interested in collisions, the slower the speed the better.
Why is that? Similar to why ‘Oumuamua didn’t hit the Sun after travelling for so long, if galaxies pass by each other at very high speeds, they don’t spend very long influencing each other. If you go to clasp hands with a friend, when you’re both walking at reasonable, low speeds, you’ll find it easy to grab onto each other and keep your hands clasped for a little while. If you imagine trying to grab onto a hand extended from a car (don’t do this), the length of time that your hands could possibly be in contact with each other is so short that at best you’re looking for a high-energy high five instead of a handshake.
Similarly, the longer two galaxies spend near each other, which they do when they’re moving slowly, there’s much more time for the two galaxies to distort each other into fantastical shapes, and the slower they go, the less energy they have in order to escape the gravitational clutches of the other galaxy. If the two galaxies are moving slowly enough, then they will sink together and scramble themselves into a single, messier, larger galaxy, keeping all the stars that had made them up before their crash together. This is the future for the Milky Way and Andromeda - while the Sun will remain in orbit around the new center of our newly enlarged galaxy, the skies will be dramatically changed.
There are lots of places in the universe where galaxies can orbit at much much faster speeds. We don’t expect them to collide in the same spectacular fashion as the Milky Way and the Andromeda galaxy do, because they’re moving so much faster. In galaxy clusters, which are home to hundreds or thousands of galaxies, the relative speeds between any two galaxies can be much, much faster - up to thousands of kilometers per second. At a thousand kilometers per second you’d reach Tokyo from New York in ten seconds flat, take six and a half minutes to get to the Moon, and make it to Pluto in a little under three months. At that speed, even if the galaxies come near each other, they're the equivalent of trying to grab your friend's hand from the window of a high speed train - over very quickly. Only a direct hit between the disks of two galaxies would cause these same kinds of streamers to appear we see from the slower collisions. Given the amount of space between galaxies, even in the relatively dense regions of richly populated clusters, that almost never happens.
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