+ is right and up.
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Case 1: Detached wall, we don't catch the balls
As you threw the balls off you and the card would move in the opposite direction under the laws of conservation of momentum. You would gain speed m(ball)/m(cart)*vball for each ball.
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Correct.
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Case 2: Detached wall, catching the ball.
You would gain the above velocity each time you threw a ball, and again each time you caught one.
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Correct.
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Case 3 Attached wall, not catching the ball.
You would move opposite the direction if your initial throws. The ball bounces off the wall and heads off in the other direction, behinid you. You will move opposite the final motion of the ball.
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Incorrect. You'd remain still.
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Case 4 Attached wall, catching the ball
After you do everything you will be again sitting on the cart with a bunch of motinoless balls. You must also be motionless.
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Incorrect, you'd move in my + direction.
