Factual error: There's absolutely no way Sallah and Indy would be able to lift that 'lid' (for lack of a better word) off the ark. That thing is solid rock and would weigh close to a metric ton - and none of them look like professional deadlifters to me.
Factual error: When Indy is attempting to steal the Ark when driving the truck, one of the soldiers shoots him in the arm. Then another German punches him repeatedly in the same arm, causing Indy great pain and injury. After suffering a wound like that there's no way he'd be able to move that arm, let alone raise it above his head. But the next day after the boat is stopped by the sub, Indy jumps into the water and swims to the sub without any problems.
Continuity mistake: When Indy stands inside the Map Room, looking at the model of the city, the rays of the sun are to his left, and they move right and down as the sun rises. But since the entry hole is behind him, that would mean that the sun would be to the right when standing outside - but it was to his left when he entered the hole.
Continuity mistake: When the temple collapses, Indy is holding the idol with his right hand against his chest, and the left hand free and arm extended. A frame later, both arms are extended and holding the idol.
Continuity mistake: When Toht frees Marion, Dietrich enters the tent, and a few seconds later, Gobler does the same. In the next shot, Gobler enters first, and Dietrich second. (01:07:52)
Continuity mistake: When Belloq and the Germans are at the Well of the Souls, a top-down view shot when Indy yells, "Why don't you come on down here and I'll show you!" shows Belloq and a German soldier kneeling down to the left of him, also looking down into the hole. But, the side view shots show only Belloq standing near the opening.
Continuity mistake: When Indy is listening to the one Army agent he is in the process of putting his right hand into his pocket. In the next shot the hand is back on the leather case. (00:16:46)
Continuity mistake: When the Cairo Henchman pulls the knife on Marion, the blade edge changes position from down to up to down again.
Revealing mistake: During the big car chase where Indy follows the ark, he comes down a hill on a horse to get into the truck. You can tell the ground has already been disturbed by the horse from previous takes.
Revealing mistake: After the German is chopped up in the propeller and the plane explodes, you can the blasting cord exploding along the path to the explosion.
Continuity mistake: When Indy sits down in the seat of the plane look behind him, there is nobody holding a Life magazine or fitting the description of the man later shown watching him from the other seat.
Continuity mistake: During the truck chase, as Indy is fighting with the German driver, they knock over some scaffolding and a man drops down in front of the windshield. None of the workers on the scaffolding landed on the truck, much less the front of it. So, he shouldn't have been there.
Continuity mistake: In Indy's class at the beginning, the girl that lowers her eye lids to show the phrase "Love you", is looking in a totally different direction than the people behind her. They're all supposed to be looking at him, but the girls behind her are looking to their right while she is looking straight. They can't be daydreaming because they're all smiling and look infatuated. (00:13:58)
Continuity mistake: When Marcus enters the classroom he leans up against a wall and there are 2 girls in the front row next to him. When the bell rings, the girls are the first two students to leave the classroom. Seconds later when Indy looks over to Marcus the same two girls are just getting up from their desks.
Continuity mistake: Indy and a Nazi soldier are fighting in the cab of the truck when they come to a place where a stone building is being built. As the truck is driving into the scaffolding, several rocks come down and hit the truck and remove the rack from the top. Moments later the rack is back in place. (02:50:00 - 03:01:00)
Continuity mistake: As the big German engages Indy in the fight, Marion is removing the chocks from one of the wheels. In the wide shot you don't see her at all, even climbing the plane to knock out of the pilot.
Other mistake: After Indy shoots the driver of the truck that Marion is supposedly in, the camera looks at the driver's bloody hand falling limp. In the shot of the truck tipping over, you see the stunt driver working the wheel.
Continuity mistake: When Indiana is confronted by the Cairo swordsman, he already has his left shoulder pointing towards his opponent in the shot where the crowd scatters. But in the following shot, where the swordsman is first seen, Indiana has his right shoulder pointing forward and is just turning around to his left side. (00:40:14)
Continuity mistake: While escaping from the boulder, right before the spiderweb scene, Indy runs facing forward and makes a right turn. A shot later from a wide angle he is facing backwards.
Continuity mistake: When Indy is driving the truck, chasing Belloq and his Nazi cohorts, the last Nazi guarding the Ark of the Covenant climbs to the driver's side of the truck and kicks Indy away from the steering wheel. The first time he punches Indy in the arm, Indy's blood is on his fist. However, the second time he punches Indy in the same spot, there is no blood on his fist.
Answer: Indy and Marcus Brody believed that the Ark needed to be studied. They certainly didn't want to put it back where it was found. They believed that the U.S. government would find the best archaeologists, researchers, and scientists in the world to study the Ark. They are both upset that instead, the government has decided to simply lock the Ark away. This is why Indy says, "Fools. They don't know what they've got there," as he is leaving the building.
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