Asteriod Question

I'm a screenwriter and I have a question:

If a big enough asteroid hit Jupiter and a piece of Jupiter broke off, could
it become a rogue asteroid and head for Earth? (I've heard that Jupiter's
many moons were once actually part of Jupiter that broke off.)

If not, what type of situation can be proposed that would produce a
threatening rogue asteroid?

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From the Desk of Lise Eleanor
http://www3.sympatico.ca/l_e/