Horizontal displacement of a falling object from trajectory length

Question

I have an object falling down in a parabolic trajectory. I can estimate the total distance traveled during the time t, i.e. the length of the parabola's arc is known. I need an efficient algorithm that estimates the horizontal displacement of an object. The estimation can be rough. An error within, say, 2-3% is acceptable.

Answer 1

Given a (differentiable) function $f: \mathbb{R} \to \mathbb{R}$, the arc length $l$ of the path between points $f(\alpha)$ and $f(\beta)$ is given by $$ l = \int_\alpha^\beta \sqrt{1+(f'(x))^2}\,dx $$

A general quadratic function is given by $f(x) = ax^2+bx+c$, and so $f'(x) = 2ax + b$. Substituting this into the expression for $l$ above and integrating gives $$ l = \left.\frac{1}{4a} \left((b+2ax) \sqrt{1+(b+2ax)^2} + \text{arcsinh}(b+2ax) \right)\right|^\beta_\alpha \ . $$ The distance traveled in the $x$-direction, say $\Delta x$, is $\beta - \alpha$. You have stated that you know $l$; in order to know $\Delta x$ you need to know at the very least $a$ and $b$ (the coefficients of the quadratic function $f$). In this event, you might be able to solve the preceding equation for $\Delta x$ (technically it would be one equation in two unknowns, so I am not even sure you could manipulate it to get $\beta - \alpha$). However, if you know $a$ and $b$ then it is likely that you can find $\Delta x$ without having to go to this trouble.