Metaphorically describes someone's attempt to make you care, often about something that matters about as much as a gram of piss. The care missile is generally depicted as a large nuclear warhead with a smiley face visible somewhere on its body. Make sure to notify the launcher when the missile doesn't hit its mark.

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See also cool story bro.
Ryan: Oh my God, dude. I ate this taco five minutes ago and I just about shat my pants.

Jim: WHOOOOSH! Hear that? That's the care missile landing in the Sea of Japan.
by Care Engineer <3 December 4, 2009
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