Momentum (of a particle) is the product of its mass times its velocity.
Period. That is how it is DEFINED. Just like the velocity is DEFINED as the distance traveled divided by the time it takes to travel it.

Mass, distance and time are physical properties. Momentum and velocity are mathematical constructs based on calculations done with and on physical properties. In relativistic physics, even mass, distance and time are relative properties, not intrinsic ones.

To put it another way, an object has a mass, it is an intrinsic property.
It does not have a velocity until you place it in a coordinate system which allows you to determine its velocity relative to those coordinates. But coordinate systems are purely arbitrary, you can draw one wherever you like. Any property that depends on where you draw the coordinate system has no physical reality of its own.

A bullet has a great deal of momentum which it imparts to its stationary target when it strikes it. To a flea riding on that bullet, it has no momentum. To another bullet flying alongside at the same speed in the same direction, there is no momentum. Without the target, the momentum of the bullet is a meaningless concept. Momentum is purely an artifact of the space-time relationship of bullet and target. It has nothing to do with the bullet or the target, it is a property of their relationship to each other.

As for Archimedes, he discovered how to calculate the mass of water displaced by an object. I.e., he discovered how to describe and determine density. This allowed him to determine the density of a complex shape which was impractical to determine by measurement. People had been building ships that floated for thousands of years before Archimedes and already understood this perfectly well. He just demonstrated the relationship was a precise and computable one and had a practical application..

"How" is a physical concept. "Why" is a psychological one. "Why" implies purpose, which implies deliberate design, motivation and planning, which implies a supernatural intelligence. A supernatural intelligence may very well exist, but we have absolutely no evidence for it, and science has shown us no need for one in our explanations of reality.

Science can answer the question of how nature works. Why it works the way it does isn't even a proper question, it's more of an artifact of language. It sort of like the "It" you are referring to in the sentence "It is going to rain today."