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Our physics teacher turns off the lights and has a tiny square of light open through the window, everything else being completely dark. There is a parking lot outside of our class and she is holding a giant white board and on it we see can see the trees (in shadow form) and also the reflections of light from the tops of cars. She goes outside of the class room and then drops her red coat on the ground. We see her walking, although in shadow form and on the ceiling ( I don't know why it worked where she was on the ceiling but she was) and when she drops her red coat, a red blob appears on the ground. Why does the red blob show when she rests it on the ground and not while she is walking. Also how does this all happen from just a tiny square of light opening from the window?

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This concept is called "Camera obscura". In short, as the hole in the wall is much smaller than objects outside, light rays get focused in the hole and then expand back out in the room, thus acting as a convex lens, producing a mirror image on the opposite wall. We can see the image because outside is brighter than inside and incoming light makes the wall much brighter than it previously was. That's why You had to turn the lights off.

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