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## Main Question or Discussion Point

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

[tex]x(t)=x_{0}+V_{0x}*t-g_{x}*t^{2}[/tex]

[tex]y(t)=y_{0}+V_{0y}*t-g_{y}*t^{2}[/tex]

[tex]z(t)=z_{0}+V_{0z}*t-g_{z}*t^{2}[/tex]

where

[tex]g_{x}=\frac{G*M}{r_{x}}[/tex]

[tex]g_{y}=\frac{G*M}{r_{y}}[/tex]

[tex]g_{z}=\frac{G*M}{r_{z}}[/tex]

and the axis projected on the r-axis

[tex]r_{x}=x*cos\theta*sin\phi[/tex]

[tex]r_{y}=y*sin\theta*sin\phi[/tex]

[tex]r_{z}=z*cos\phi[/tex]

After introducing [tex]\theta[/tex] and [tex]\phi[/tex] the whole thing becomes difficult. Is there an easier way?

[tex]x(t)=x_{0}+V_{0x}*t-g_{x}*t^{2}[/tex]

[tex]y(t)=y_{0}+V_{0y}*t-g_{y}*t^{2}[/tex]

[tex]z(t)=z_{0}+V_{0z}*t-g_{z}*t^{2}[/tex]

where

[tex]g_{x}=\frac{G*M}{r_{x}}[/tex]

[tex]g_{y}=\frac{G*M}{r_{y}}[/tex]

[tex]g_{z}=\frac{G*M}{r_{z}}[/tex]

and the axis projected on the r-axis

[tex]r_{x}=x*cos\theta*sin\phi[/tex]

[tex]r_{y}=y*sin\theta*sin\phi[/tex]

[tex]r_{z}=z*cos\phi[/tex]

After introducing [tex]\theta[/tex] and [tex]\phi[/tex] the whole thing becomes difficult. Is there an easier way?