This is part of a personal project... I've recently become addicted to modeling various physical systems from scratch, such that I find explicit solutions of position as a function of time, and graph em.(adsbygoogle = window.adsbygoogle || []).push({});

But I've hit a glass ceiling trying to find an analytic solution to the 1-dimensional problem free-fall problem where acceleration due to gravity is the inverse square of the distance:

$$

a = \frac{-1}{x^2}

$$

To keep it as simple as possible, I'm assuming the initial velocity is zero, and the initial position is positive. With a little differential manipulation I found:

$$

v = -\sqrt{\frac{2}{x}-\frac{2}{x_i}}

$$

I'm 99% confident that's correct where ##0<x\leq x_i##, and it seems intuitive. It follows from that, then:

$$

t = \int \frac{-1}{\sqrt{\frac{2}{x}-\frac{2}{x_i}}}\,dx

$$

Now this is a nasty integral, but itdoeshave an explicit solution. You just cannot explicitly solve for x, but I'm perfectly happy with t in terms of x rather than x in terms of t (Because I can just convert it to a parametric form).

The only problem is: Wolfram_Alpha and MATLAB disagree on what the integral is, and I have no idea what to do with the constant, besides. Also, Wolfram_Alpha returns an expression with complex values, which I don't understand or want.

If anybody knows how to do this, any help would be vastly appreciated... I've only formally taken up to Calculus II, and I've been struggling with this for weeks now. (I fear I simply don't have the required math knowledge yet... e.g. what is a Lagrangian?)

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# Solving nonlinear differential to emulate Newtonian gravity

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