Gravitation and Motion Equations

[Sobeita]

The standard equation for motion is:

X= .5 * a_x * t^2 + v_x * t (+ c_x)
Y= .5 * a_y * t^2 + v_y * t (+ c_y)

...and of course, you can expand it to include more axes and more orders if you like.

Gravitation is a force, but to use it in the motion equation, you need it in acceleration form.

F_g= G * mM / r^2;
F= m*a; Ag = Fg/m;
A_g= G * M / r^2.

But if you use substitution, using A_g, the motion equation is still always parabolic, despite what we know about orbits. The cause is obviously the presence of the "r" variable, which is dependent on the position of the object (so that acceleration changes based on its position, and that position affects the acceleration, and both change continuously.)

With an iterative approach, I am able to approximate motion due to gravitation. The result is essentially a long string of parabolas.

Is there a way to combine these two equations, A_g and motion, into a true unified equation?

 

[Doc Al]

The standard equation for motion is:

X= .5 * a_x * t^2 + v_x * t (+ c_x)
Y= .5 * a_y * t^2 + v_y * t (+ c_y)

These equations only apply to motion with constant acceleration; they are not true in general.

 

[Sobeita]

That was more or less the purpose of posting this question. Is there an equation that can calculate motion due to gravitation, without using some sort of sigma iteration approach?