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As we already know that the concept of Momentum was discovered before Newton discovered his laws of motion, but my question is $\rightarrow$ How they discovered the relationship p=mv without knowing much about the concept of Force at that time, and while performing the experiments don't the objects on which they are performing experiments always starts accelerating. How they maintain the constant velocity to found the relationship p=mv?

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I already made this comment back in P.SE before you copied it here, but you seem not to have taken heed:

without knowing much about the concept of Force at that time

The concept of forces, all the way to constructing vectors as a mathematical system allowing addition of vectors and resolving of forces, was done before even Galileo, let alone Newton. One thus cannot say that people before Newton didn't know about forces.

In particular, Stevin's textbook with the "Epitaph of Stevinus" on its cover details the above.


Back onto topic, the thing you are looking for is documented even on Wikipedia. You can trace the genesis of the concept, first of something that might persist for a while in 6th century, John Philoponus's writings, to Arabic writings in 11th century, Avicenna (Ibn Sīnā)'s, asserting that they would persist indefinitely in a vacuum.

I am less happy than the those works, than the sudden appearance in 14th century by Jean Buridan who defined $\text{impetus}=\text{weight}\times\text{velocity}$, which is then basically Newtonian.

Newton had a better framework, because he was able to describe gravity correctly, and thus define the concept of mass and weight as separate things, and so that is some improvement, but other than that, the concept of impetus of Buridan is already everything that is really needed. The leap the Newton needed, is thus very much smaller than one would expect.

https://en.wikipedia.org/wiki/Theory_of_impetus

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