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Last updated: Fri, Jun 30, 2017
(This is not a physics lesson.)
It's very important when you work with models to be careful about definitions. The equation p1 = p2 has a very particular meaning. It says that momentum is preserved (the momentum before is equal to the momentum after), but it is true only in a particular class of situations. For instance, the bodies must form what is called a "closed system," and if the bodies collide, each of the bodies must be "ideally elastic."
"Well," you may say, "what is momentum in the first place?" The answer is that p = mv. (Momentum equals mass times velocity.) Each of the elements of this model (in this case, the elements are the terms of the equations) is defined very precisely. "Mass" has a very particular definition, as has "velocity." Mass is the property of matter that determines the gravitational attraction between two bodies. It is measured by determining the gravitational attraction between the object of interest and a known second object, usually the earth, in which case the mass is also the weight.
The standard unit of mass is defined in the SI (metric) system by the international prototype kilogram, which is stored at the International Bureau of Weights and Measures in Paris. To determine the mass of an object, you compare it to the mass of the international prototype kilogram. (You do it indirectly. Working copies of the prototype mass have been made and used to calibrate weighing instruments, which in turn are used to calibrate others, and so forth, until finally you or I can easily determine the approximate mass of an object by setting it on top of a scale.)
An immense amount of effort has gone into developing these definitions and the relationships between them. Nowadays an engineer can use the equations to design pin-ball machines, provided he/she bears in mind the definitions along with the conditions under which they are accurate predictors of real-world behavior.
An "operational definition" is the definition of something in terms of the procedure by which it is observed, classified, or measured. Operational definitions are a critical part of science because they turn concepts (like "mass" or "weight") into concrete procedures (“weigh it on a calibrated scale at sea level on the equator if you want a very accurate answer”). Although we can manipulate ideas in our heads (Einstein famously formulated relativity this way), operationalizing a concept allows it to be tested. It allows others to participate not just in formulating and testing ideas, but eventually by applying the model.
Results from mechanical physics are so well-established now that they seem self-evident, but the science that I've described in the last few paragraphs took literally millennia to develop. Prior to the 17th century when Galileo made his mechanical investigations, people had explanations for mechanical phenonmenon. An object in flight, they might say, continued in flight because of the vortices it made in the surrounding air. This theory is good enough if you're philosophizing, but it would predict that an object wouldn't have momentum in space.
Consider this: How would you define pain? This is a big problem in pain research. In a sense the most important thing about pain is that it hurts. But this is not observable or comparable across individuals, so how could you perform research on it? Perhaps you could define it in terms of people's reports of it ("on a scale from 0 to 10"), but then you're no longer talking about pain itself, you're talking about people's answers to a question. How does that affect what you know about pain itself?