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Momentum is a key concept in physics that describes the amount of motion an object has. It is a product of the object's mass and velocity and is usually represented as \( p = mv \).
Momentum is a vector quantity, meaning it has both magnitude and direction. This is important because in any physical process, both the magnitude and the direction of momentum can affect the outcome.

When an object experiences a force and changes its velocity, its momentum changes. In problems involving multiple objects, such as the one with the vase, the total momentum of the system before an interaction must equal the total momentum after the interaction, provided no external forces act on the system. This principle is known as the conservation of momentum.

In the context of physics, a collision occurs when two or more objects exert forces on each other for a relatively short period of time.
Collisions can be perfectly elastic, perfectly inelastic, or somewhere in between. In elastic collisions, both kinetic energy and momentum are conserved. In inelastic collisions, only momentum is conserved, which often results in some deformation or generation of heat.

For this problem, involving a vase shattering, it's important to understand how the fragments move post-collision. Even though it wasn't a traditional collision between objects, the fragments behave similarly post-breakage. They separate with different velocities but still respect the laws of momentum conservation.

Vector components break down forces or motion into different directions, typically using an x, y, and sometimes z-axis in physics problems.
In the problem of the vase breaking, you're dealing with two main vector components: along the x-axis and y-axis. The pieces of the vase move independently in their respective directions.

To fully describe the motion of each piece, you calculate the momentum in these two directions separately. This involves breaking vectors into horizontal (\( x \)) and vertical (\( y \)) components, ensuring each component's momentum sums to zero along its direction per the conservation laws. Later, these components are combined again to find the resultant motion or speed of an object, like the third piece here.

A frictionless surface is an idealized concept used to simplify physics problems. It means that there is no resistance to motion, allowing objects to move freely without any energy loss due to friction.
This simplification is crucial in momentum conservation problems because it eliminates external forces that could alter momentum. Without friction, any changes in an object's speed or path must result from interactions with other objects, not from energy loss to the surroundings.

For the falling and breaking vase, assuming a frictionless surface means that the total momentum before the vase hits the floor is conserved after breaking. It allows the problem to be analyzed using only internal interactions, which simplifies calculations and focuses solely on momentum principles.