Why does a person float rather than staying in the elevator's floor in a free fall? Many people have asked online why does a person float on a free falling elevator in earth. But my question is why doesn't a person just stay in the elevator's floor when the elevator is in a free fall? Why do they float like this:


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*Albert Einstein's Elevator Thought Experiment

*Zero G Flight
The elevator is falling with an acceleration of $g=9.8$ because of earth's gravity. But the person will also be accelerating with the same rate (neglecting air resistance). So shouldn't the person be in the same place (the elevator's perspective) from the elevator's perspective?
 A: When we analyse the situation from a freely falling elevator, an non-inertial frame of reference, we need to apply inertial (pseudo or fictious) force in order to apply the familiar Newton's laws of motion. Here, the pseudo force acts in the upward direction and cancels the weight of the person. The net force on the person is zero. 
The following diagram shows how the inertial force and the gravitational force on an object in a freely falling elevator cancel each other out, thereby leading to weightlessness:

Image source: My own work :)
Now let's imagine that you're in an elevator of a very tall building. When the cables of the elevator are intact, you'll be standing on the elevator's floor and the floor would exert normal contact force on you. When the cables are cut, the elevator accelerates downward with acceleration equal to the local $g$ (assuming no friction with the rails and zero air resistance). Your feet will still remain in contact with the floor. However, you'll no longer feel the upward force exerted by the floor on your feet. If you lift your legs up, you'll not move down opposed to what happens normally.
I find the first video of Albert Einstein in the elevator to be quite accurate. Once the elevator starts its free fall, he doesn't spin randomly. He stays upright and when he lifts his legs, he just looses contact with the floor. However his centre of mass remains in the same position (with respect to the lift) if we assume he doesn't kick the floor or the walls of the elevator.
The second video in the question is not from a freely falling elevator however. It's taken in a reduced gravity aircraft. These aircrafts follows the following trajectory to simulate reduced or zero gravity environments:

Image source: Purdue University
The reason why the people rise above in the clip linked by you is, the aircraft initially accelerates downward more than the local value of $g$. You can experience a similar situation in an elevator, if a giant monster pushes the elevator down so hard that it accelerates down with an acceleration greater than $g$ say $50~\mathrm{ms^{-2}}$. In this case, with respect to the elevator, you'll not only be moving upward, but will be accelerating upward with an acceleration equal to $a-g$ where $a$ is the acceleration provided by the monster. Now you can stand on the elevator's roof!

If you're interested in how such aircrafts stimulate reduced gravity environments, I've given a short explanation in this answer.
A: Yes, they would stay in the same place... Except for the fact that from the elevator's frame any slight push upwards will propel you and you can't stop that... Like if you were in a space station. In the second example you gave you can clearly see that all the people push upwards which causes them to float.. if you stayed perfectly still you would still be on the floor.
A: While other answers are not wrong, I'd like to add that the main reason why someone's feet can't stay on the ground in free fall is that any time there is a slight pressure or bump between feet and floor due to body movement, the contact force will push the feet upwards. Since no other forces are involved, the feet will keep floating away from the floor.
Additionally, legs act like slightly compressed springs : when free fall starts, the muscles that were still tensing to counter gravity will keep pushing the floor, so the reaction force of the floor will keep pushing the feet up, even though it's weaker and weaker as the legs "extend". By the time the forces reach zero because the legs are no longer "compressed", the center of gravity of your body has acquired some speed relative to the floor so you float away.
Note that in the first example, Einstein's body should have had a speed relative to the elevator, since his feet were already far from the floor, resulting in his head bumping in the ceiling. Which we can confirm by looking at the second video (but the plane isn't perfectly in free fall obviously so relative speed can vary a lot).
