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To get into the sun, we’d probably want to fuel the rockets in space using reaction material mined in space (from the moon or an asteroid). That would more or less eliminate the problem you’re talking about, which is why I kind of skipped over that in my comment. But you’re right; this is one of a million things that makes space travel hard and expensive.
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We can get up to any speed with enough time and fuel. The trash rockets would just need to get into a solar orbit, and then burn retrograde for a fairly long while. Or if you add a gravity assist in, this is doable today; the Parker Solar Probe got to (and indeed beyond) that speed, for instance. It’s easier and quicker when there aren’t squishy people aboard (we don’t tend to like acceleration much higher than 9.8m/s², for instance).
Both can be true: it is too expensive, and there’s no money to be made. $840B wouldn’t put a dent in the launch costs for the tens of thousands of rockets we’d need to put into space over the next several decades in order to just get rid of the Pacific Garbage Patch, to say nothing of the rest of the trash on this planet.
And actually, there’s a third true thing: it wouldn’t help much. Having it on Earth isn’t the problem; it’s having it in the oceans that’s the problem. Partially because of the environmental impact, partially because of the biological impact, and partially because we don’t have access to it to reuse it, so we have to keep making more. Once we had it out of the oceans, we could recycle it or even just sequester it away.