Factual error: Enterprise and Vengeance come out of warp near the moon, 237,000km from Earth. Due to their altercation. They lose power and proceed to fall freely toward Earth. The scene plays as though it takes a matter of a few minutes, too fast to get the situation under control. At that distance, the gravity acceleration from Earth would only be .01 m/sec/sec. This means that they should have had approximately 2.52 days before crashing, especially given their apparent relative stop as per the visible moon.

Star Trek Into Darkness (2013)
Trailers
Directed by: J.J. Abrams
Starring: Benedict Cumberbatch, Simon Pegg, Chris Pine, Karl Urban, Zoe Saldana, Anton Yelchin, Alice Eve, John Cho, Zachary Quinto
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Christopher Pike: Are you giving me attitude, Spock?
Spock: I'm expressing multiple attitudes simultaneously, Sir. To which are you referring?
Trivia: As Kirk, Spock, and Uhura head to the shuttle bay for their mission to Kronos to capture Khan, Sulu orders crew to prepare the vessel that was confiscated in the Mudd incident. This is a reference to Harry Mudd, a roguish character who appeared in the original Star Trek series in the episodes titled,"I, Mudd" and "Mudd's Women." Harry Mudd, played by Roger C. Carmel, was the only non-regular character to appear in more than one episode on the original Star Trek series. Carmel was slated to reprise the role in Star Trek: TNG, but he died before the episode could be filmed. (00:44:25)





Answer: Unless the cold fusion went directly into the volcano, all the way to the source of the pressure.