Function for orbits based on time.

[RandomMystery]

Okay, I am trying to find a formula for the position of two bodies (or more) in space based on the initial velocity and time.

I'm trying to integrate the equation for gravitational acceleration to find the velocity equation, however, the radius is changing, and the degree (theta) is also changing:

Edit:Does anyone know any calculus II and above method to solve this? I can't so far with Calc I

 

[HallsofIvy]

You will be able to reduce that a bit by using "quadrature". Let v= dy/dt. Then
$$\frac{d^2y}{dt^2}= \frac{d}{dt}\left(\frac{dy}{dt}\right)= \frac{dv}{dt}$$
and, by the chain rule,
$$\frac{dv}{dt}= \frac{dv}{dy}\frac{dy}{dt}= v\frac{dv}{dt}$$

so you have
$$\frac{dv}{dy}= \frac{Gmy}{\sqrt[3/2]{x^2+ y^2}}$$
Of course, you will have to have something of the same form in terms of x, probably, letting
u= dx/dt
$$\frac{du}{dx}= \frac{Gmx}{\sqrt[3/2]{x^2+ y^2}}$$


I suspect that, at best, you will be able to reduce that to an elliptic integral (so named because they arise in calculating the elliptic orbits of planets) which cannot be integrated in terms of elementary functions.