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There's a block slides downward along a frictionless wedge which sits on a frictionless horizontal surface, when the block leaves the wedge, both of the wedge and the block have a horizontal velocity in opposite direction.

In this situation,can we say the momentum of the system is conserved? How to deal with the vertical motion of the block? If I say the system is conserved, should the system include earth? Since the force of gravity of the block as an external force of the block-wedge system has done work during the motion.

Any idea or hint is appreciated!

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    $\begingroup$ Only the horizontal momentum is conserved, since there is a vertical external force. You can include Earth so that all momentum components are conserved, but it's not helpful for solving the problem (i.e. you get 1 more equation to use, but also 1 more variable). $\endgroup$ – knzhou Apr 13 '16 at 1:28
  • $\begingroup$ so if I have the horizontal velocity of both of the block and wedge I can say m1v1 = m2v2? (the system initially is at rest) @knzhou $\endgroup$ – JTing Apr 13 '16 at 1:30
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Yes and no. depends on the direction considered. Momentum is a vector quantity. Since there is no force in the x direction, momentum is conserved in that direction but not in the vertical direction because gravity (an external force for the system of Wedge and block) is acting.

Yes, for your 2nd comment above. You can use COLM (conservation of linear momentum)

To analyse the motion of the system taking into account the vertical direction AND the horizontal direction, you would need to use COLM equation and the Energy conservation theorem as everything is frictionless.

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