Revealing mistake: While Indy and Marion are in the Well of Souls, and they encounter the snakes, Indy falls to the ground only to be confronted by a hissing Cobra rearing its head. Look carefully and you'll see the snake's reflection on the safety glass between it and Indy. Briefly you can also see the torch's reflection while he's waving it around. You can also see Marion's reflection when she falls in later on and is face to face with the cobra. [This has been corrected in the new DVD set. You can only see it if you have the original VHS. I think it's visible in the "making of" on the DVD as well.]
Revealing mistake: Watch the shadow of the stone Indy pushes out to escape from the Well Of Souls - it bounces down the sand dune, exactly the way a large rock shouldn't. (01:15:30)
Revealing mistake: After Indy gets behind the wheel of the truck containing the ark, he jerks the steering wheel to the right to run the pursuing jeep off the road. He then jerks the wheel to the left to hit an upcoming motorcycle. In neither case do you see his body sway with the presumably drastic swerve of the truck. Apparently this was filmed in a stationary vehicle, or one in tow that was fitted with a prop steering wheel.
Revealing mistake: When Indy fights the driver of the Nazi truck, he gets thrown through the windshield, and eventually ends up underneath the truck. When he is climbing underneath the truck, heading towards the back of it, you can see the trench that was dug down the middle of the road to provide the stuntman (this being one of the few stunts that Ford didn't do himself) with the extra room needed to do this feat. (This is backed up by DVD commentary. The director stated that the truck was too low to the ground so they had to dig the trench in order for there to be enough room for the stuntman). (01:27:50)
Revealing mistake: When Indy jumps across the pit to pursue the spy that took the idol, he barely makes it across, grabbing the ledge on the opposite side with his hands. When he pulls himself up, he kicks loose a rock and the entire "solid" rock wall shakes like it's nothing more than a foam wall.
Revealing mistake: Where is all the light coming from in the Well of Souls, even after the entrance was re-sealed, and the only torch still burning went out?
Revealing mistake: Just before Belloq says "beautiful", look carefully at the woman's ghost in the center of the screen. You can see one of the wires on her back that makes the stunt woman fly, a black line above her left shoulder.
Revealing mistake: When Indy is fighting the big German at the plane, if you look at the wheels when the plane is turning around, you can see a chain and sprocket assembly that is driving the wheel to actually turn the plane. (01:18:55)
Revealing mistake: After Marion goes in the basket, there is a chase and a truck tips over, if you look amidst the dust, you can see the ram used to put the vehicle on its side.
Revealing mistake: When the German staff car comes into the market place to find no truck, watch as Dietrich hurls the melon - the SS guy in the front is obviously a dummy for this one shot. (01:29:40)
Revealing mistake: When Indy and Marion are being sealed up in the Well of Souls, You can plainly see Marion counting down (she is mouthing the numbers) to the cue for her to scream as the Well is sealed. (01:12:20)
Revealing mistake: When Indy fights the driver of the Nazi truck, the windshield is smashed when Indy is thrown through it, but there is no wind. Look at the driver's hair, not one stirs from a breeze while driving a fast moving truck with no windshield? (01:27:30)
Revealing mistake: When Indy is running through the cave and finds the spy that took the artifact with spikes through him, he brushes up against the dummy and it shakes like rubber and the light bounces off it like rubber. (00:09:10)
Revealing mistake: The Pan American aeroplane Indiana Jones boards in San Fransisco looks similar to (but has noticeable differences from) the Boeing 314 PanAm used for transpacific flights. This aeroplane did however not fly until 1938 and at the time of the story the company only flew transpacific with Sikorsky S-42s, an aeroplane with a completely different body than the one used in the scene.
Revealing mistake: When Indy is fighting the big German at the base of the airplane, Indy reaches down and grabs a handful of sand to throw in his face. You can see the finger marks in the sand from a previous take.
Revealing mistake: During the truck chase scene, Indiana is thrown or forced over the front of the bonnet and almost run over. There is a wide camera shot, and while holding on to the front fender, Indiana can be seen sitting on a chair which is attached to the truck. (01:27:40)
Revealing mistake: When we have the exterior shot of the ship that Harrison Ford, Karen Allen, and the ark are aboard, even on the DVD can be seen specks of dirt over the view of the ship and the ocean - obviously it was filmed through the window of another ship. (01:31:35)
Revealing mistake: When Belloq's head explodes, the actor has been visibly replaced by a dummy.
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.
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.
BaconIsMyBFF