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In contrast to a popular question, here I want to hit a ceiling of an elevator. If the elevator is standing still, my legs are not strong enough to generate a jump that will help me hitting the ceiling. At the same time, if the elevator is in a free fall, my friend claims that it becomes easier to hit a ceiling if I jump. I am convinced though that my maximal location above the elevator's floor after a jump is invariant no matter whether the elevator is standing still or falling. Who's right here and why?

I suspect that he may be right, since in case the elevator starts accelerating down beyond $g$, I won't even need to jump in order to hit the elevator, and continuity argument should make my friend right.

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  • $\begingroup$ In the elevator's frame, you are not accelerating. It is much like the situation in a space station. $\endgroup$
    – Sid
    Commented Aug 17, 2023 at 9:44

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In free fall, in theory any push whatsoever on the floor of the elevator will get you to the ceiling.

The elevator is falling the same way as you are, so there is no gravitational pull pulling you down to its floor. Thus it is as if you are floating in the international space station, where you merely push a surface and propel yourself and continue in motion until you reach another surface.

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Straight from Wikipedia:

In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it. https://www.wikiwand.com/en/Free_fall

Now let's think in terms of a system that has no force acting on it. Then, the only thing that is accelerating is your action of pushing you of the floor of the elevator. Hence, you would require infinitesimal work assuming no friction. Which is way less than you would need to reach the elevator top if you would in a "still standing" elevator. Therefore, it would be "easier".

Another argument without using that free fall behaves as if no force is acting (again assuming no friction) is, that if you're accelerating exactly the same as the elevator you would require only a small change in velocity to eventually reach the top. You can think of it in this way:

Suppose you're falling with the elevator (same speed) and at some point in time you decide to jump you therefore create a velocity upwards for yourself. Ignoring that you would also accelerate the elevator downwards (which would also help you to reach the top), you would get something like this: $$ v_{\text{Person}} = -gt + \delta v,\\ v_{\text{ElevatorTop}} = -gt $$ With $v$ being the respective velocity, then your relative velocity to the elevator would be $r_\text{rel} = \delta v$. You would therefore reach the elevator top after some time. Again the velocity you would have to generate would be infinitesimal.

When the elevator is standing still, you essentially have (after jumping): $$ v_{\text{Person}} = -gt + \delta v,\\ v_{\text{ElevatorTop}} = 0 $$ It would require you to generate a velocity that is big enough to counteract the acceleration to get $r_\text{rel} > 0$ for enough time to reach the elevator (depends on distance of floor to top), which requires more energy.

Hope that helps.

Cheers

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If the lift is accelerating downwards with an acceleration $a$ and you drop a object inside the lift its acceleration relative to the lift will be $g-a$.

In free fall $a=g$ and so an object will not accelerate relative to the lift, it appears to be weightless to an observer in the lift.

Thus, even the smallest initial upward velocity of the object will allow it to reach the ceiling.

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You and the elevator are in free fall both accelerating downward at g. When you “jump”you exert a force on the elevator floor and the elevator floor exerts a force on you. During this “jump” you have imparted downward momentum to the elevator and simultaneously imparted upward momentum to yourself. This leads to a discrepancy in the acceleration of you and the elevator during the “jump”. During the jump the elevator briefly accelerated downward more than g and you briefly accelerated downward at less than g this caused the elevator to gain an increase in velocity compared to you. So after the “jump” both you and the elevator return to accelerating downward at g however the now elevator has that additional velocity. So you will inevitably hit the elevator ceiling. How quickly this happens depends on factors such as the force and duration of your jump and the mass of both you and the elevator.

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