[MrWookie47]

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Edit:In the brief time between when you release the ball and when the ball hits the wall, you travel in the -x direction.

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Thats not true, unless you are standing on roller skates or something. When you throw the ball in the +x direction you do not have any momentum in the -x direction because that momentum is absorbed by the friction between your feet and whatever you are standing on.

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I believe we're standing on the cart, and said cart has perfect bearings. So, yes, we do move.