The True Nature of Gravity

Note: the original posts were removed, because they became unnecessary for this thread after I worked out a solution. Special thanks to Lozmaster for helping out with the preliminaries.

I wanted to derive an equation that can model objects falling from over long distances, where the acceleration due to gravity is not assumed to be constant. In other words, the difference in a(t) is not negligible. The reason I wanted to solve this equation is that I was taught motion due to gravity is parabolic, and I've known it isn't due to Newton's law. At first I tried finding the solution by looking online and in textbooks, but I couldn't find anything. Eventually, I gave up and attempted the problem myself.




















Making the following substitutions makes the equation cleaner.







Starting here, I use a different notation so that I can use the separation of variables method to integrate the function.







Substituting u as an operator allows for use of trigonometric integrals.











For the following substitutions, draw a triangle.

















I integrated from 0 to t. I used this notation, because it is impossible for me to find θs that correspond to r(0) and r(t). I wanted to find the functions that correspond to these substitutions before plugging back 0 and t.



Notice that the constant C on the left side is due to the r(0) term. Notice that the definite integral on the right side yields no second term.





I substituted back from the triangle I drew earlier.







I substituted sqrt® back in the place of the u operator.



Now that r is back, I can determine the constant term, which is basically just plugging in 0 into the function r. I noted this as r(0). I noted r(t) as just r, because r(t) is already implied by r.









I changed the variables so that they are less annoying to deal with. I chose h for initial height, r(0), and v for initial velocity, r'(0).







I'm going to simplify the asinh term first:



Plugging back in:

















This is just about the most simplified form I could get the above equation to. I like this form because when r=h, it's immediately apparent that t=0 and the ln term must subtract from the first part of t®.

Alternatively, the equation can be written as the following:



Where:



The equation gives the exact trajectory of the falling object when graphed. Albeit, I was unable to write r in terms of t, but it's easy to graph the function and just change the axes.

At a glance, the ln term will approximately cancel out when the difference between r and h is negligible. Without the ln term, the motion is exactly parabolic.

When the equation is graphed, it appears that the motion is parabolic over small distances, but over large distances, the time it actually takes for an object to fall through increasing acceleration is about half that of the time it actually takes for an object to fall through constant acceleration.


Someone please check my work. Much scrutiny is appreciated.