Solve 2-Body Equation for t w/ Given Parameters

[Philosophaie]

For a 2 Body Equation:

$$x = - \frac{1}{2} \frac{GM}{r^2}cos(\theta) cos(\phi) t^2 +v_x t + x_0$$
$$y= - \frac{1}{2} \frac{GM}{r^2} sin(\theta) cos(\phi) t^2 +v_y t + y_0$$
$$z= - \frac{1}{2} \frac{GM}{r^2} sin(\phi) t^2 +v_z t + z_0$$


$$r= sqrt(x^2 + y^2 + z^2)$$
$$\theta = atan(\frac{y}{x})$$
$$\phi = acos(\frac{z}{r})$$

Given:$$v_x, v_y, v_z, x_0, y_0, z_0 and M.$$

Now all I have to solve for t.

 

[SteamKing]

I see a bunch of formulas. What do they represent?

 

[Philosophaie]

I am working in Cartesian Coordinates. I want to solve for x, y, z and t. I want to do this in Newtonian Physics.

For the fourth equation in the four equations and four unknowns I choose the geodesic:

$$(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2 = 0$$

Are the geodesic, the x equation, the y equation, and the z equation (above) correct?

 

[Simon Bridge]

I am working in Cartesian Coordinates. I want to solve for x, y, z and t. I want to do this in Newtonian Physics.

You want to solve for x,y,z, and t, for what?

Are the geodesic, the x equation, the y equation, and the z equation (above) correct?

It is not possible to tell since you won't tell us what these equations are supposed to represent. What system are you attempting to model?

I suspect that the equations are not correct.

 

[PJC01]

I would first like to clarify that the given parameters and equations represent the motion of two bodies in a gravitational field, where M represents the mass of the central body, and x, y, and z represent the positions of the two bodies in three-dimensional space.

To solve for t, we can use the equations for x, y, and z to eliminate the trigonometric functions and solve for t. This can be done by setting the equations equal to each other and then rearranging them in terms of t. The resulting equation will be a quadratic equation, which can then be solved using the quadratic formula.

It is important to note that the solution for t will have two values, representing the two possible times at which the two bodies will have the same position in space. This is known as the time of closest approach or the time of collision.

Additionally, it is worth mentioning that the given equations assume the two bodies are point masses and do not take into account any external forces or perturbations. Therefore, the solution for t may not accurately represent the actual motion of the two bodies in a real-world scenario.

In conclusion, the given parameters and equations can be used to solve for t, which represents the time at which the two bodies will have the same position in space. However, it is important to consider the limitations and assumptions of these equations in order to accurately interpret the results.