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A thought experiment has been bothering me. Assume an enclosed box free-falling in a constant gravity field. The box contains a person and a bowling ball at the top end. The person throws the bowling ball down (in the direction the system is falling/the direction gravity is pulling). Obviously, being a closed system, forces are all internal, so there's no net effect in that sense. But, the center of mass has shifted downward some amount more than what is due to gravity.

The question, then, is this: What does someone outside the box observe? Part of me says they don't see anything, the box appears to fall under the influence of gravity with no visible change. The other part says there's a period corresponding to the flight of the bowling ball where the box falls faster than can be accounted for by gravity.

A follow-up question that I believe is closely related is the following: Imagine jumping off a high-dive board in a cannonball position. Shortly before hitting the water, you extend your legs downward. Do you hit the water faster, slower, or with the same velocity as someone who kept their legs extended throughout the jump?

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If you are initially pressed up against the top of the free-falling box when you throw the bowling ball, then, the center of mass will indeed be lowered (in your frame). Now, as you mentioned, the system is enclosed and all forces are internal other than gravity. So, the center of mass of the system must follow the free-fall pathway. In order for this to occur while the CoM moves down, the box must actually slow down temporarily. Then, when the CoM stops moving relative to the box, it must speed up again.

Here's how this happens. When you throw the ball, from Newton's 3rd law, we know that there must be a force of equal magnitude being applied to you, and thus to the wall upon which you are pressed. The impulse (integrated force over the time during which it acts) from the throw slows down the box since it is directed opposite to the direction of travel. While the ball is in flight, the CoM moves at the original free-fall velocity and thus moves downward relative to the box. Then, when the ball hits the opposite side of the box, it is stopped. The impulse from stopping the ball returns the box to its original speed, with the new CoM at the correct location on the free-fall flight path.

Now, if you start away from the wall, then nothing will be seen until you (who are thrown backwards upon tossing the ball) hit the wall. Then the above scenario occurs. In the meantime (between tossing and hitting the wall), the CoM remains fixed relative to the box since all forces so far are internal to the you-ball system.

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