Did Newton conduct any experiments to find something called momentum, or was he such a great genius that he was able to spot it intuitively?
K=1
The value of K was arbitrarily defined. Actually as momentum was to be defined here, Newton had a freedom of choosing any constant except 0. He could have chosen it to be 2 but that would make the calculations difficult as compared to K = 1. This is what gives you the definition of momentum as mass times velocity. In case Newton would have chosen it to be 2, your definition would have been twice of mass times velocity.
Experiment
Actually Newton's second law is (F)(∆t) ∝ (m)(v). This was what was experimentally found by Newton by changing one of the parameters while the others remained fixed.
He used
- Meter scale
- Stopwatch
- Glass Surfaces to reduce friction
Conservation of momentum
This quantity of (m)(v) was defined as a new thing called Momentum. Now if F remains zero then (m)(v) doesn't change and hence momentum remains conserved.