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If people hit the ceiling, then the plane has to have moved down faster than the he people. Doesn't that imply the plane moved downwards faster than gravity?
Yes, it's the plane being accelerated in a downward direction by something. Most of the time that's turbulence, but it appears there is more investigation ongoing here as the dip was much more severe than usual.
Yes
Turbulence can do funny shit with aerodynamic lift. However, I think most of the injuries are from people lightly hitting the ceiling from slight upward momentum (kind of like when your car goes over a big bump and you float up out of your seat) and then getting slammed to their seat when lift is very suddenly restored.
Yes it does. The fact that it moved faster than gravity is not special in and of itself - many things do that when acted upon by other forces (e.g. if you thrown a ball towards the ground, and in a similar case mentioned by another commenter). In the case of the plane though, the question is why? Was it a failure of the avionics package, erroneous input by the pilot, etc. Knowing the cause will help understand and mitigate the risks moving forward.
Sometimes, you see this phenomenon when throwing a frisbee. It can rapidly move up and down when it's travelling a long distance due to physics and shit, probably something to do with hitting pockets of warmer and cooler air on the field. Planes can do that on a massive scale, and you better be buckled when it happens.