Significance of time being dependent in many equations for gravity? I have noticed, when browsing the equations for a falling body, it is often the case that distance is the independent variable rather than time. Take for instance, the time $t$ taken for an object to fall from a height $r$ to a height $x$,
$$t=\frac{\arcsin\left(\sqrt{\frac{x}{r}}\right)+\sqrt{\frac{x}{r}\left(1-\frac{x}{r}\right)}}{\sqrt{2G(m_1+m_2)}}.$$
Of course, indeed we are trying solving for $t$ here, however note that this equation is also the solution to the differential equation made by the question: "What distance $x$ has an object fallen from a height $r$ to, after a duration $t$?" In either case, $t$ is not the independent variable (that is, $x$ cannot be solved for using elementary functions, whereas $t$ can). This is one example of many that arise in solving differential equations involving gravity.
Although I am a math student, and not a physics student, I cannot help but be mystified by this possibly insignificant occurrence. In my dominantly classical understanding of physics, it would certainly seem like time should be the independent variable here. As far as I'm aware, measures of distance in physical interactions are determined by the time elapsed , rather than the other way around.

Is this observation a significant insight into the nature of time and space (possibly resolved by non-classical understanding), or is the non-independence of time in these equations just a consequence of our mathematics?

 A: You've identified a real phenomenon, but it doesn't have a deep meaning; it's just a consequence of how we solve these equations. One-dimensional motion in a potential $U(x)$ is governed by Newton's second law,
$$F = m\frac{d^2 x}{dt^2}, \quad F = - \frac{dU}{dx}.$$
However, in general, solving second-order differential equations is very difficult. So we usually instead use energy conservation,
$$\frac12 mv^2 + U(x) = E$$
in order to get an equivalent first order differential equation,
$$\frac{dx}{dt} = v = \sqrt{\frac{2 (E - U(x))}{m}}.$$
The solution to this differential equation is straightforward,
$$\int dt = \int dx \, \sqrt{\frac{m}{2 (E - U(x))}}$$
and performing the integral automatically gives $t$ in terms of $x$. This sort of thing happens constantly in mechanics. Sometimes you can invert the relation to write $x$ in terms of $t$. Other times you can't, but you can get a numeric solution.
As a general point, when describing something as simple as one-dimensional motion, it simply just doesn't matter to physicists whether $x$, $v$, or $t$ is being considered the "independent variable" at the moment. They're just all physical quantities that vary together. Try asking a lion running at you what independent variable they're using in their heads to determine how they move! For that matter, ask them if they're taking care to run in a way described by an elementary function. A lot of the niceties people pay attention to in math classes are irrelevant baggage when describing reality, which do nothing but weigh down the user's mind.
