Gravitational Path of an Object

[Philosophaie]

What is the path of an object entering the graviational pull starting at a point $$[x_{0}, y_{0}, z_{0}]$$ with a velocity $$[V_{0x}, V_{0y}, V_{0z}]$$ neglecting air resistance? This is what I have thus far:

$$x(t)=x_{0}+V_{0x}*t-g_{x}*t^{2}$$
$$y(t)=y_{0}+V_{0y}*t-g_{y}*t^{2}$$
$$z(t)=z_{0}+V_{0z}*t-g_{z}*t^{2}$$

where
$$g_{x}=\frac{G*M}{r_{x}}$$
$$g_{y}=\frac{G*M}{r_{y}}$$
$$g_{z}=\frac{G*M}{r_{z}}$$

and the axis projected on the r-axis
$$r_{x}=x*cos\theta*sin\phi$$
$$r_{y}=y*sin\theta*sin\phi$$
$$r_{z}=z*cos\phi$$

After introducing $$\theta$$ and $$\phi$$ the whole thing becomes difficult. Is there an easier way?

 

[D H]

What is the path of an object entering the graviational pull starting at a point $$[x_{0}, y_{0}, z_{0}]$$ with a velocity $$[V_{0x}, V_{0y}, V_{0z}]$$ neglecting air resistance? This is what I have thus far:

$$x(t)=x_{0}+V_{0x}*t-g_{x}*t^{2}$$
$$y(t)=y_{0}+V_{0y}*t-g_{y}*t^{2}$$
$$z(t)=z_{0}+V_{0z}*t-g_{z}*t^{2}$$

where
$$g_{x}=\frac{G*M}{r_{x}}$$
$$g_{y}=\frac{G*M}{r_{y}}$$
$$g_{z}=\frac{G*M}{r_{z}}$$

NO!

Philosophaie, based on your other posts, you have a marked tendency to apply equations randomly and incorrectly. Correcting these equations would be a disservice to you because you not understand the theory. Without this understanding, you might use the right equation this time, but you will use the wrong equations again in the future. Please, read a book. Here are three:

Bate, Mueller, White, "Fundamentals of Astrodynamics". [URL]https://www.amazon.com/dp/0486600610/?tag=pfamazon01-20[/URL][/URL]
Cost at Amazon: $16.61

Vallado, "Fundamentals of Astrodynamics and Applications". [URL]https://www.amazon.com/dp/1881883140/?tag=pfamazon01-20[/URL]
Cost at Amazon: $63.95.

Roy, "Orbital Motion". [URL]https://www.amazon.com/dp/0852742290/?tag=pfamazon01-20[/URL]
Cost at Amazon: $70.00.

Cost of these at a library: Free.