Final answer:
Momentum in the horizontal direction is conserved when no external horizontal forces, such as air resistance or friction, act on a system. Momentum is a vector and can be conserved separately along the x, y, and z dimensions. In the absence of horizontal forces, the x-component of momentum remains unchanged over time.
Step-by-step explanation:
It is justified to say that the momentum in the horizontal direction is conserved because, typically in a system where there's no air resistance or any horizontal forces acting, there is no net force in the horizontal direction. According to Newton's first law, an object in motion will stay in motion at the same speed and direction unless acted upon by a net external force.
When looking at projectile motion, or situations where surfaces are frictionless, there are no horizontal forces acting on an object. Hence, the horizontal momentum — the product of mass and velocity in the x-direction — remains constant.
Furthermore, momentum is a vector quantity, meaning it has both magnitude and direction, and it can be broken down into independent x, y, and z components. Since forces act independently in these perpendicular directions, the conservation of momentum can be applied separately along each axis. In the absence of external forces in the horizontal direction, the x-component of momentum does not change over time, satisfying the principle of conservation of momentum.
In contrast, in the vertical direction, gravity acts downwards, which is why the momentum in that direction is not conserved — unless considering the momentum of a closed system like the projectile-Earth system as a whole where the internal forces (like gravity) do not change the total momentum.