You ask an incredibly good question! I should start by saying that it's really hard to change these ideas into mathematical formulas and it is by no means a simple process.

When a physicist considers how to build a mathematical formula, they need to consider how each variable varies with respect to the others. In some cases, this can be tested out with experiments.

For example, if we consider the equation for kinetic energy (E=(1/2)m*v², where E is energy, m is mass, and v is velocity), one could design an experiment where you measure the energy based on a series of tests where you either only vary the velocity or the mass of the projectile. Then, by looking at your data, you could see if there was a quadratic dependence (i.e. x²) or a linear dependence (i.e. x) on each varied parameter. In other cases, the physicist has to put forth a hypothesis for the equation and follow the math until there are no mistakes or they run into a conflict. At that point, it is almost like an experiment, but instead of designing a physical test of the hypothesis, the physicist uses math to examine what their hypothesis implies and how that stacks up against well established theories.

Finally, a physicist can understand that two parameters are proportional (i.e. E is proportional to m*v²), but still not know the constants that go into that equation. Constants have to be derived from experiments. For some constants that are known to many digits, very difficult and accurate measurements are required to ensure that enough significant figures can be obtained. I hope that helps!

Here is another exploration of the methods used as well: learn here.