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When an elevator suddenly starts decelerating downwards after the cable snaps, why does the person inside the elevator float upwards? In such a scenario why aren't the elevator floor and the person always in contact because the only force that is acting on both objects is gravity so should they not be accelerating at the same rate downwards; therefore, the distance between them should be zero right?

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    $\begingroup$ You are neglecting elastic compression in the shoes and legs and whatnot between the person and the elevator. See this possible duplicate question. $\endgroup$
    – rob
    Mar 28 at 4:22

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If one had an object such as a steel box in the elevator, it would remain on the floor. In free fall, all objects would fall at the same rate, so the box would fall at the same rate as the elevator, and thus remain at the same location relative to the elevator.

People, however, are a more complicated story. A surprising amount of energy is stored in the tendons of the leg while standing upright. When the elevator falls, and both bodies enter free fall, the energy in those tendons is released. The result is that the person is propelled "upward" with respect to the elevator. If a person was capable of standing upright without loading tendons in this way, they would stay on the floor of the elevator, just like the box did.

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  • $\begingroup$ I just realised. If we consider the steel box inside an elevator when the cable snaps scenario, you stated that the distance between them would be zero as they fall at the same rate. Does this mean they are still exerting a normal force upon each other? $\endgroup$ Mar 31 at 1:24
  • $\begingroup$ I am asking because at the instant of the cable snapping, they would still be exerting a normal force upon each other; therefore, the upwards normal force upon the steel box would cause it to have a lesser acceleration before it reaches 9.8 in comparison to the elevator. As the lift has the downward force of gravity and the normal force the box exerts upon it, which causes the elevator's acceleration to be initially greater than 9.8. This increase initial acceleration of the elevator would cause it to have a greater velocity in comparison to the box; thus, creating distance between them right? $\endgroup$ Mar 31 at 1:26
  • $\begingroup$ I had just started learning Physics, so please be patient while answering if you believe certain aspects of my understanding is ridiculously wrong. $\endgroup$ Mar 31 at 1:28
  • $\begingroup$ In the simplified version, the cable snaps instantaniously, going from supporting the elevator to supporting nothing in 0 time. It's discontinuous. In that case, the moment the cable snaps, the normal forces go to zero. The elevator is now in free fall. In reality, these things happen over a (very) short period of time, and there's a ton more details. But I do think they end up only confusing the issue. I don't know if you have learned calculus, but that mathematics handles such discontinuities and their associated infinitesimally short periods of time very rigorously. $\endgroup$
    – Cort Ammon
    Mar 31 at 3:02
  • $\begingroup$ (In the real world, there's elastic materials stretching, tearing cables deforming, friction, and all sorts of fun effects. But really the only value of considering them at this point is if the discontinuitiy really bugs you) $\endgroup$
    – Cort Ammon
    Mar 31 at 3:04
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Assuming the elevator begins accelerating downwards at acceleration $g$, in the frame of the elevator the person is weightless, i.e. has zero acceleration. This reflects the fact that the person and elevator are accelerating at the same rate. For the remainder of this discussion, let's place ourselves in the frame of the elevator.

In this ideal scenario, in the absence of any other external forces, it's just like a person is in a stationary box floating in outer space. The person won't move relative to the elevator, so they can freely rotate about their center of mass. I suppose if they wanted to they can bring their legs up from the floor of the elevator, or propel themselves by pushing off the walls/floor of the elevator. Note that any contact force with the ground will result in the person accelerating up off the ground, so it's easy for them to start floating upwards.

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