During the ITV News "UFO Landing Confirmed" breaking news scene, the lower caption shows the text "Amateur footage confirms rumors". ITV News is a British news network and would actually spell 'rumors' as 'rumours'.
When the alien communications array is recovered, someone says it's made up of material which "isn't on the periodic table." All possible chemical elements can be placed on the periodic table. The array may have been made up of an "as yet undiscovered element", but no scientist would have said an element was not on the periodic table.
When Stone falls after the impact of the glass, a cut appears on the left side of his his face. When he gets up, it moves further to the left. In subsequent shots it moves to the right side, then continually changes sides.
As wreckage from an alien craft crashes on Hong Kong Island the view shows a large seated Buddha located on the mountains above the city. This is actually located at Po Lin monastery which is on Lantau Island some distance from the area actually shown (and is well inland and in a largely rural area, unlike the water-front urban area shown in the shot).
The boilers on the Missouri are shown to generate steam pressure instantaneously, so the ship can be moved free of its moorings. A real Iowa-class battleship's boilers need 9 to 14 hours to generate enough steam pressure to move the ship.
When the full reverse command was given, an underwater view showed the propellers reversing their rotation. Arleigh Burke destroyers utilize reversible pitch propellers and therefore do not change their rotational direction during reversing maneuvers, they simply change the pitch of the blades to provide reversing thrust.
During the ceremony at the end of the film, Admiral Shane presents the Navy Cross to Lieutenant Commander Hopper on behalf of his older brother, who was killed in combat. Hopper is not a Lieutenant Commander; he wears the shoulder boards of a Lieutenant, and he is not promoted for his actions in the film.
The anchor cable is secured to a ship via a cable clench located deep in the cable locker. Detaching it requires some serious human effort, not the automated release implied in the movie by the order "release the anchors".
Many shipmates call Chief Petty Officer Walter "The Beast" Lynch as "Beast". It is not proper Navy protocol for subordinates or officers to address him by a nickname. While on duty, they should address him as either Chief or Chief Lynch.
The crew of John Paul Jones readies the USS Missouri, a floating museum, for combat within an hour or two, albeit with the help of retired battleship veterans. In the early 1980s, reactivating and modernizing Iowa-class battleships took two years.
The alien ships are shown moving through space, crossing Jupiter's orbit and arriving at Earth in minutes. Earth and Jupiter are about one light-hour apart, so they are moving faster than light, which is impossible as nothing can *move* faster than light. Other means of traveling from one place to another could include teleportation, but the ships are shown literally moving.
The main engines of Arleigh Burke destroyers are not in the open as was shown in the engine room. They are compartmentalized in a housing that is big enough to allow maintenance and repairs, but also allows the tremendous heat produced by the engine to be ventilated and also provides isolation for fire suppression.
In the beginning, a signal is sent from Earth, via a satellite, in a laser-like form into deep space, targeted at the nearest Earth-like planet that has been discovered. Somehow, the signal travels the distance in the blink at an eye, despite the fact that no beam or wave can travel faster than light, meaning that it would take the signal over four years to reach even Proxima Centauri, the closest star to the Sun, let alone some deep, deep space planet.
Before the port anchor is dropped, the guns are rotated to "210". The guns are later shown to be aimed to starboard, and when the ship turns to port, the guns are facing the alien ship. "210" refers to 210 degrees, which translates to just left of directly backwards, or 7 O'clock. For guns trained to 210 to be aimed at the alien ship, the Missouri would have had to turn almost completely around to the right before firing.
When the Royal Australian Air Force (RAAF) jets save the Battleship, the camera cuts to a view within the cockpit, and a flag is visible on the pilot's helmet. The flag is intended to be the RAAF ensign, but the RAAF roundel is missing from the bottom right corner, making it a mis-colored Australian flag.
On the John Paul Jones, the CO and the XO are killed while standing on the bridge wing when the alien round goes off, making Hopper the next in line for command. Standard procedure in a battle situation would be to spread out the leadership just so this wouldn't occur. Typically the CO would be on the bridge, and the XO would be in CIC or vice versa. Additionally, the third in command on a Navy destroyer is typically the chief engineer. Hopper, as a fire control officer, would not be third in command after the CO of the XO.
When the alien ship fires an ultrasonic attack, all the glass windows on the destroyer break. Once the ultrasonic sound stops, some crew members use binoculars with intact lenses. They should've broken along with the windows.
The goof item below may give away important plot points.
When the JPJ sinks it goes down by the bow. In the next scene the crew abandons ship. When captain Nagata and Hopper are "climbing" the deck to reach the stern, one shot reveals the position of the ship being near vertical. Still the two "walk" towards the stern, defying gravity.