# If a person drops a briefcase in an elevator and it does not fall to the floor, what is the elevator's aceleration?

I read this question on my Physics book and I'm still wondering whether my answer is right. My first thought is that the elevator is accelerating downwards. If it were accelerating upwards the briefcase would fall to the floor even quicker than if the elevator was not accelerating at all.

Then, the question is, how much is it accelerating? My answer is $$a \ge g$$. If the acceleration of the elevator is the same as that due to gravity, the elevator and the briefcase will experience the same acceleration, and the briefcase will never touch the floor. If, on the other hand, the acceleration of the elevator is greater than the acceleration of the suitcase, then the elevator is moving faster than the suitcase and the suitcase will eventually touch the elevator's roof.

Is my reasoning correct? I found a good explanation online, but they claim $$a = g$$ is the only right answer. Am I right to think that $$a > g$$ is plausible too?

• "the briefcase will never touch the floor" On a practical note, I would say that at some point in time, the briefcase will always touch the elevator floor :) Nov 22 '21 at 10:43

When answering these sort of questions you do not just have to understand the physics, you also have to make a model of the examiner's mind. Yes, if your accelerate downwards a rate greater than $$g$$ the briefcase will not hit the floor. At $$a>g$$ (downwards positive) it will hit the ceiling. Quite possibly the question creator did not think of this possibility. They are human, and possibly overworked, tired, and underpaid. Have mercy on them unless this is a critical exam for you.

• Thank you so much for the answer! Of course I understand people overlooking this. As a matter of fact, the site where I got the answer is unrelated to the book I got the question from. And the explanation they gave for the problem is quite detailed. The site normally gives me great answers about Math and Physics. So of course I understand they overlooking this. After all, they know way more about Physics than I do! Nov 22 '21 at 3:22
• As I see it the question does not contradict the possibility of a > g. Nov 22 '21 at 9:01

If the briefcase remains in equilibrium with the elevator, then the elevator is accelerating downwards. This means that $$a=-g$$, where the minus sign means the motion is downwards.

If the briefcase rises to the elevator's ceiling, then $$a \geq -g$$.

• Thank you for the answer! The literal question from the book reads as follows: "A woman in an elevator lets go of her briefcase, but it does not fall to the floor. How is the elevator moving?". Therefore I think it's reasonable to consider the case it does not touch the floor, but it touches the ceiling, like you did! Nov 22 '21 at 3:24
• If the briefcase rises to the ceiling then $a<-g$. Nov 22 '21 at 4:28
• @DanielFBest I think you messed up, $a=-g$ means $a$ is opposite in direction to $g$, if I am not wrong. So $a\geq g$, or $a=g$ depending on what you consider the situation to be Nov 22 '21 at 7:15
• @Java Monke Often $g$ is to be taken the absolute value of the averaged gravitational acceleration (take it as a scalar quantity). Adding the minus sign makes it a vectorial quantity (albeit 1-D in this case). Nov 22 '21 at 9:24
• What about air resistance? In the scenario in the question, I assume one can consider the elevator to be hermetic. Should a need to be slightly less than g to achieve equilibrium considering air resistance?
– d-b
Nov 22 '21 at 10:48

Well, with a>g, the ceiling is the new floor, so it still drops to the floor, in a manner of speaking. With a=g, it does not drop at all. So the answer is "impossible scenario" ;-).

• I think the book only means the floor below the persons feet as the only "floor", according to the what we generally consider a "floor" Nov 22 '21 at 19:32
• @JavaMonke Yes, so the answer depends on your ability to maintain a head stand ;-). Nov 22 '21 at 20:43
• Thank you for the answer. That actually crossed my mind haha. But I'm sure when they say "floor" they mean where the person is standing. Nov 22 '21 at 20:56
• @JosuéFuentes One may assume "the surface on which they were standing when they entered", or "the surface with the PVC on it", or "the surface without the light and the vent" ;-). The answer is in the spirit of Einstein who, of course, came up with the elevator gedankenexperiment, and famously came to the conclusion that you could not tell what's "up" and "down", or more generally, that "acceleration" is indistinguishable from "gravity", or even more generally, that all spacetime curvatures are created equal. Nov 23 '21 at 6:47