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During the close-up of Fish watching Palmer's Larry King interview, the TV screen is reflected in Fish's glasses. In that reflected image Mr. King is talking, but the audio from the TV is Palmer's voice.
Ellie says, "There are over four hundred billion stars out there, just in our galaxy alone. If just one in a million of those have planets, and if just one of a million of those have life, and if just one in a million of those have intelligent life, there would be literally millions of civilizations out there." The math doesn't work. 400,000,000,000 times (1/1000000) times (1/1000000) times (1/1000000) is far less than one. The movie's creators were informed of this issue after the scene was shot and couldn't find a cost effective way of fixing the problem, so they left it in.
When the pods drop through the machine commences, the ship containing the control staff rocks severely (causing some of the control staff members to shout in panic as they lose their balance), and the machine itself is releasing brilliant light and a thunderous roar as the rings whirl like helicopter blades. Three or four seconds later, when the pod splashes into the catch net, there is little ambient noise. In the ship, staff members are suddenly sure-footed and tranquil.
Crew or equipment visible
Errors in geography
The opening sequence contain impossible physics with the four planets (Earth, Mars, Jupiter and Saturn) that appear completely desynchronized as the sequence zooms out. The Great Red Spot on the gas planet Jupiter appear larger and centered near the equator rather than slightly positioned at the middle lower line of the planet.
Every scene outside supposedly NASA facilities repeatedly mentioned as being in Florida show the California state flag flying below the US Flag. The official state flag of Florida contains a red-bar cross with the Florida state seal in the middle. The state flag of California shows a bear in the middle with a red bar at the bottom. The two flags could not be more different. The Florida state flag should be flying in those scenes, not California.
The suggestion that a man-made Earth-orbiting satellite could have been used to simulate a signal emanating from a star is absurd. A satellite within Earth's or the Sun's gravitational fields can not carry enough fuel to maintain a position in front of a distant star for more than a fraction of a second out of every several hours. And since the signal was tracked from several locations on Earth, the sky would have to be filled with hoax satellites to fool more than one listening station. Moreover, the VLA alone would be able to confirm the distance of the signal from parallax.
The VLA in New Mexico does not listen for radio transmissions, but in fact takes radio "photographs" of space. In their Visitor Center they acknowledge the filming of Contact (1997) but debunk the premise of the film that their dishes could be utilized for the purpose described.
When Ellie runs down the stairs to check on her father, she turns left to go into the room where her father lays dying. When the shot changes to Ellie running back up the stairs to get his medicine from the bathroom cabinet, the view is flipped all the way up the stairs, down the hallway, and into the bathroom. As Ellie reaches for the medicine cabinet, we now see she was in a "mirror world" (where everything would be flipped). This was an effective camera trick used for the audience to help add to the chaos, confusion, and fear that Ellie was suddenly feeling as her world was crashing down.
Incorrectly regarded as goofs
When giving testimony at the inquiry, Ellie states that the travel lasted 18 hours for her and just a second for the witnesses. She had no way of knowing the exact duration of her experience since she didn't have a watch and any scientist who admitted being subject to a vivid illusion would know that subjective time is hard to reckon. Besides, the duration of the static recording was unknown to her and the public.
If an alien race wished to transmit a signal consisting of prime numbers far across the Galaxy, it should be sufficiently advanced to realize that AM is a very inefficient mode of transmission. It would be far better for the aliens to use a single carrier wave (CW) using either simple on-off keying (like Morse code), or frequency shift keying (FSK). FSK would give the transmitted signal an extra 3dB in power over simple CW.
After Ellie notices the compass is floating smoothly, she unstraps from the chair inside the "IPV" and floats as well. Soon, the chair shakes violently loose from its anchor point, then slams into one side of the pod. When Mission Control resumes video contact with her, she's lying on the floor of the IPV, her cheek bloodied, but it's hard to tell if the chair is still mounted or not. The condition/position of the chair would confirm or refute her story of having traveled, rather than dropping straight through the spinning rings.