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.
Revealing 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.
Revealing mistake: As the truck tips to the left, the stunt driver is seen steering, and appears to be activating the ram to finish the stunt.
Revealing mistake: When the thug with the blade sticking out his back is standing still, there is some kind of squared device on his back under his clothes, presumably to retract the fake blade to end the sequence.
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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