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I have long wondered what would happen if - theoretically - someone was in an elevator that was moving rather quickly and either jumping or flying a helicopter (something of the like).

So here's the situation, the elevator is moving upwards in a building and going at a fairly high rate of speed. The occupant of the elevator jumps up while the car is still in motion. If the room had been still, they would reach a "high point" of, say, 12" and then begin falling. However, since the car is moving, will that still be the case, will they still fall 12" or will it be less than that because the car is moving upwards?

I know this is probably a dumb question, but I can't picture how it would work!

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If you stand still and jump, you exert a force on the floor. This force gives an acceleration (from $\sum F=ma$) that brings you to a certain height. From this height you fall back the same distance down to the floor again.

If the floor under you is moving upwards, you can jump as always and exert the same force as always to reach the same height (measured from the point where you jumped from) as always. But when falling down from this height, the floor comes up to you while you fall towards it. So you reach the floor higher up than you left it.

So, you jump just as high, but fall less.

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    $\begingroup$ Your last sentence uses two different frames of reference in only six words! $\endgroup$ – DJohnM Aug 31 '16 at 21:11
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If the elevator is not accelerating, there is no way for the person in the elevator to know whether it's moving or still. So the laws of physics will be exactly the same: they jump with a certain force, reach a certain height, and land again.

If the elevator is accelerating (upwards), then the person will feel heavy: they will have difficulty jumping as high as usual (just as it was easy for astronauts on the moon to jump really high, as gravity was less there). So for the same "extra" force, they will appear to rise less high in the elevator. This is of course because, from an external observer's point of view, the elevator is accelerating up towards the passenger. But for the passenger it just feels like somebody "boosted gravity".

The reverse happens if the elevator is accelerating down (in the limiting case, the elevator could be in free fall). Now the passenger feels lighter, and can jump higher than usual. Until the free fall case, where the passenger will float (briefly, depending on the height of the building).

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    $\begingroup$ +1 it's an appropriate question to ask today, after this incident m.rte.ie/news/2016/0831/813120-turbulence-shannon-landing 14 people standing in the plane bounced off the roof of the aircraft, it fell 1,000 feet in clear air turbulence. $\endgroup$ – user108787 Aug 31 '16 at 21:16
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With respect to an observer in the elevator, they will fall down the same distance, 12" since the velocity of the moving elevator adds up to the initial velocity of jumping (assuming the elevator is not accelerating).

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