By trial and error, and experimentation.

For a simple example, if you know that higher air pressure means more friction, and a lower terminal velocity, then you would have to find a way to slap in these three values in a way so that they affect each other in just the right way.

You write as a tentative formula, like “Terminal Velocy = Mass / Air Density”, and then you do a bunch of tests to see if it gives you sensical values. Then, you go so close, that the next step is to use it to predict stuff, and refine the finer details.

Science is about 90% trial & error, and 10% random freakin’ luck.

There have been lots of people throughout history. Some of them were pretty smart

They generally start with something(s) they already knew that was simpler and either expanded upon it, generalized it, or combined it with something else that they knew or saw. They then tested it to see how well it work

There were a couple Newtons/Einsteins who completely turned everything on its head, but that’s the exceptionally rare case

Technology helps validate theories. Those theories are developed today through the same means and methods used for the last thousand years.

We’re better observers now, thanks to technology, but all of that was reliant on the countless iterations of formulae and theory tested and retested over generations.

We do something called mathematical modeling – look at some subset of the universe that we want to explain, try to find mathematical rules that match up with how it behaves, and test whether those mathematical rules actually match up with a set of experiments that are designed to challenge it.

The modern form of this activity was started by Newton, though arguably it goes back to Aristotle. Newton noticed that more massive objects were more difficult to get moving, but if they were put on something very smooth, once you got them moving they would pretty much keep moving. So he hypothesized the simplest relationship that matches this behavior. The rate at which an object’s velocity changes is inversely proportional to its mass, and the relationship between them is the force applied. In other words, a = F / m, or acceleration (rate at which velocity is changing) is equal to the force applied, divided by the mass of the object. This is usually taught as F = ma.

This accurately described the simplified system of massive objects on very smooth surfaces. It turns out, if you include all the complicated effects we find in nature, like friction, this simple law actually predicts a huge amount of the motion we see. This is the essence of mathematical modeling – you simplify the natural world down to a simple rule, and then see how far that simple rule can take you.

These descriptions get more complicated over time because they are built on top of one another. Newton noticed that if he assumed his force law was responsible for planetary motion, it looked like objects in space pulled on each other with a force that got weaker as you got farther away. There are many possible mathematical forms of such a force, but he luckily had another fact – Kepler’s laws, which made certain geometric statements about orbits. It turns out that the combination of F = ma and those geometric statements mean that only one force law actually worked – the inverse square law, F = G m1 m2 / r^(2). This is pretty complicated, but the steps to get to it were kind of simple. We noticed that all orbits had a certain shape, and that we had a relationship between forces and motion that seemed to work in a lot of circumstances. This is extremely useful in mathematical modeling – we can build models on top of other models, and use that to make quite complex predictions in a sequence of smaller steps.

This has been going on for hundreds of years, and the end result is that theoretical physics talks about highly abstract mathematical objects, guessing formulas based on the possible behaviors of those objects. But as always, each step is small – use a little bit of physical observation, and the rules we already have, to get possible new rules.

We still do the guessing part without technology, but the other part (seeing the consequences of our guessed rules) now often does use technology. In the past we were limited to only those predictions we could calculate by hand. It would have been very difficult, for example, to do any chaos theory without modern computers, because the behaviors of chaos theory are only really visible when you make very computationally intensive calculations. Newton had more than enough math to notice chaotic behavior, but not enough computational power.

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