Star Trek Into DarknessGoofs
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Errors in geography
When the Enterprise comes out of warp, Sulu states that they are 237,000 Km above the Earth. After Khan's vessel explodes, the Enterprise loses power and begins falling. Even assuming full gravity for so high an altitude, an object would take almost two hours to fall to ground level (Ok, call it 1 hour, 50 minutes to break up in atmosphere.) On screen, this took about 3 minutes.
As Kirk is trying to realign the warp core, Sulu declares that if they don't get the core back online, the Enterprise will burn up on reentry. When the core is realigned and power restored, the Enterprise is already within the earth's atmosphere, as evidenced by the clouds. The ship did not, in fact, burn up upon entering the atmosphere.
Incorrectly regarded as goofs
If the Vengeance and Enterprise were both traveling at Warp, the Vengeance shouldn't have been able to fire on them with phasers. Photon torpedoes are stated to travel faster than light but not phaser beams. But according to warp travel theory, ships travel within a warp bubble. Therefore, space is traveling faster than light around the ship, but within the warp bubble, the ship is static...not traveling at all. Therefore, since the Vengeance waits until she is extremely close to the Enterprise before firing phasers, it's possible that the two warp bubbles have overlapped. Thus, both ships are static in the same space, therefore the phasers would work as shown.
When the shuttle with Kirk and Spock on board arrives at the Enterprise, the camera angle makes the ship's secondary hull look perhaps twice as wide as the shuttle. But in multiple shots throughout the movie, the interior of the engineering section is considerably larger, so the camera must have been much closer to the shuttle than to the Enterprise.
Even if no one else could figure out how to use Scotty's trans-warp beaming equation to beam Khan back from Qo'noS (given that neither planet is at warp anyway), the Klingons could certainly beam a larger force into the abandoned city on their home planet, and any transporter on Earth could have assisted the Enterprise when it was near the Moon.
(at around 1h 24 mins) While planning the space jump, Sulu's display incorrectly labels the Enterprise as NCC/0514, which is the registry for the USS Kelvin from Star Trek (2009). It should read NCC/1701.