Eli5: Why are exponents so common in scientific formulas? Why is it so common that numbers in nature are multiplied by themselves?
In: Mathematics
Two main reasons that exponents come up are because something depends on the rate of change of something else, or on some kind of geometry.
In the case of uniform acceleration, the constant acceleration is the rate of change of the velocity over time. So the velocity depends on time. The velocity is the rate of change of the position over time, so the position picks up a further time dependence and behaves as t².
Geometry gives us things like the inverse square law. Imagine all of the light emitted by the sun in all directions. It expands outward in a spherical shell that increases in area as it gets farther away from the sun. Area increases as distance squared, so the intensity of the light *decreases* as distance squared because the light is evenly spread out over that expanding area. The same geometry applies to electrostatic and gravitational forces, which also fall off as 1/r².
When the rate of change of something is proportional to *itself*, you get an exponential relationship. You can see this in unconstrained population growth, e.g. of bacteria, where the number of new bacteria depends on the number that were there before and able to reproduce. You also see this in half-life behavior of radioactivity, where the number of particles decaying is proportional to how many particles you have, resulting in exponential decay.
In an oscillating spring or a pendulum, the acceleration (force) depends on the position, with a minus sign. You can translate that into a differential equation (an equation involving both a quantity and its rate of change), and the relationship you get out of that is a sine wave.
So you can get all kinds of mathematical relationships between quantities, depending on how they’re related physically.
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