Analysis of a jar falling with a fly inside

Question

This is related to other questions that are similar, but it's somewhat different. I was discussing the following situation with my father, and we don't quite agree on the result.

The situation is this: you have a jar, closed, falling due to gravity with no air resistance. Inside the jar, full of air, there's a fly, falling with the jar, located at the bottom. While the jar is falling, the fly lifts and goes to the top of the jar. The question is: does the jar reach the ground after, before or at the same time as if the fly hadn't moved?

My father thinks that the answer is this: you take the jar+air+fly as your system. The only external force acting on the system is gravity, so the trajectory of the center of mass follows the trajectory imposed by a=-g; the fly moves "up" respect to the center of mass, so it gets higher. In order to have that the trajectory is not altered, that lifting of the center of mass must result on the system "accelerating" and hitting the ground EARLIER than if the fly hadn't moved.

Me, I think there's something fishy about that "acceleration" in order to keep the trajectory invariant, because it's caused, somehow, by internal forces.

What do you think?

Thanks.

PS: editing to add the "homework and exercises" tag. It's not homework I have, but the question can be thought as belonging there. Thanks.

There's some related questions like: https://physics.stackexchange.com/q/265021/, but they are not the same. Sorry if it's a duplicate. — user3301482, Nov 05 '17 at 15:24
Possible duplicate of Acceleration of a Free-Falling System with Center-of-Mass Change — sammy gerbil, Nov 07 '17 at 16:21

score 2 · Accepted Answer · answered Nov 05 '17 at 15:45

Your father has the salient points right. The center of mass of the fly+jar+air system will fall with acceleration $a=-g$, because the only important external force is gravity.

Given that fact, if the fly moves up to the top of the jar, the bottom of the jar must move slightly downward relative to the center of mass. The system does not undergo additional acceleration (by which I mean, the center of mass acceleration stays constant), but the individual constituents of the system (the fly+jar+air) do.

The general idea is that the wings of the fly will exert a force on the air inside the jar. This will impart momentum to the air, which will be transmitted to the jar as the air flows and collides with the walls.

The situation is actually quite complicated if you look closely, but we have no need to get into the nitty-gritty details of fluid mechanics - the absolute conservation of momentum tells us everything that we need to know.

Analysis of a jar falling with a fly inside

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