Gravitational Path of an Object

[Philosophaie]

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?

 

[D H]

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]

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.

 

[charng]

I would like to clarify that the path of an object entering the gravitational pull is determined by the interplay between the object's initial velocity, the force of gravity, and the object's position in space. The equations you have provided are correct in describing the motion of an object neglecting air resistance, but they do not fully capture the complexity of the situation.

To better understand the path of an object, we can break it down into two components: the horizontal motion along the x and y axes, and the vertical motion along the z axis. The horizontal motion is determined by the initial velocity and the force of gravity in the x and y directions, as shown in your equations. However, the vertical motion is also influenced by the object's initial velocity in the z direction, as well as the gravitational force in the z direction.

Additionally, the equations you have provided assume a constant gravitational force, which is not always the case. The force of gravity depends on the distance between the object and the center of mass of the attracting body, as well as the mass of the attracting body. This means that as the object moves closer or further away from the attracting body, the force of gravity will change, and thus the path of the object will also change.

To simplify the equations, we can make some assumptions, such as considering a point mass for the attracting body and assuming a constant gravitational force. However, to fully understand the path of an object entering the gravitational pull, we must take into account all the variables and use advanced mathematical techniques such as calculus and differential equations.

In conclusion, while your equations provide a good starting point for understanding the path of an object entering the gravitational pull, it is important to remember that the situation is more complex and requires a more comprehensive approach.