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If we imagine standing on a scale inside an elevator, the scale will project our weight. As the elevator starts to move downwards we will not start to move downwards immediately; because of inertia we stay stationary for a very short period of time and then start to accelerate downwards, which causes the scale to project a lower weight. But then, because elevators are not let into free fall, we will catch up with the elevator, since gravity accelerates is downwards. When we have caught up with the elevator, what will the projected mass be? And what if the elevator is in free fall? I would think that while the elevator accelerates we would have a lower weight as our acceleration relative to the elevator is less than one g. But when the elevator is traveling at a constant speed, I would assume that when we have caught up, the scale will project the weight we had when the elevator was standing still, since our relative velocity is one g.

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You are correct , once an elevator stops accelerating and moves at a constant speed, then we feel our normal weight. If an elevator were in freefall and there was no air resistance to slow it, then we would be in freefall inside it. If there were air resistance on the elevator in freefall, it's acceleration would slow until it reached terminal velocity, where it's weight would equal it's air resistance, Once it reached terminal velocity then we would feel our normal weight inside, until it hit the bottom.

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