Solar system moves across the galaxy with 200km/s means that you have to achieve a higher speed to escape Milky Way, namely about 500km/s. Escape velocity from a solar system can similarly be estimated by velocity of object within the solar system, it’s about 40 km/s. If you manage to achieve 0 velocity relative to Sun, you will collide with it pretty quickly. Similarly if you stop in the reference frame of the Milky Way, you will move to it’s center and collide with something there.
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