For a block on a movable wedge which is released from the top of the wedge. Why can we say that the momentum is conserved in the horizontal direction? I actually know the linear momentum conservation principle but cannot identify it, please help me to get an intuitive idea of external forces acting on this system and why momentum cannot be conserved in the vertical direction.
There is only one external force (assuming friction and air drag are absent) on the system (block and wedge) here — the normal force between the wedge and the ground. And that is acting vertically upwards ("normal" force) on the wedge.
No external horizontal force —> horizontal component of the momentum of the system (wedge and block) is conserved/constant.
There is an external force in the vertical direction —> $p_y$ is not conserved.