I know what "the bullet whizzed past" means, but I'm not sure about its grammar or syntax, that is, what kind of element in the sentence 'past' is.
Is it the abbreviation of the full sentence, the bullet whizzed, being past?
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The verb WHIZZ takes Locative Complements (as opposed to Objects and so forth). These are nearly always preposition phrases. The word past here is a preposition phrase functioning as a Locative Complement. If we try to use WHIZZ without a Locative Complement, it will sound strange and ungrammatical:
Here is WHIZZ with some different Locative Complements: