The horizontal component of velocity remains constant when an object moves up or down a slanted surface because there is no external force acting horizontally to change its velocity.
When an object moves along a slanted surface, it experiences two main forces: gravitational force and the normal force. The gravitational force acts vertically downward, perpendicular to the surface, and can be resolved into two components: one parallel to the surface (the component affecting motion along the surface) and one perpendicular to the surface (the component affecting motion perpendicular to the surface).
The normal force acts perpendicular to the surface and counteracts the component of gravitational force perpendicular to the surface. It prevents the object from sinking into the surface or passing through it.
Since there is no external force acting horizontally (parallel to the surface), the net force in the horizontal direction is zero. According to Newton's first law of motion (the law of inertia), an object will maintain its velocity (including the horizontal component) unless acted upon by an external force. Therefore, the horizontal component of velocity remains constant as long as no external force is present.
In summary, as long as there are no horizontal forces acting on the object (such as friction), the object's horizontal velocity will remain constant when it moves up or down a slanted surface.