Is it possible for something to have true zero movement?

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Can something even exist without energy? And if it can will it truly not move?To be clear I know E=mc2 but can something exist without any movement?
 
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To answer your question, we need to figure out the distinction between "not moving" and "truly not moving". I think it would be best for you to spend some time thinking about your question, and then repost something clearer.
 

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