Free-falling body and Kepler's laws

In this paper, we derive an equation for the distance covered by a free-falling body as a function of time valid for any arbitrary distance. The equation, interestingly, yields exactly the time according to the Kepler's third law in the limiting case of very large distance compared to the radius of the Earth. The equation, of course, reduces to the familiar one of the distance proportional to the square of the time in the limiting case of the distance being small compared to the radius of the Earth. This could be an interesting way to introduce some aspects of the Kepler's laws to high school and undergraduate college courses through the problem of free-falling bodies without having to solve the Kepler's problem in its entirety.