Gravity in the x, y and z directions

1. Aug 20, 2013

Philosophaie

I am working in classical mechanics. A planet is orbiting a star. The planet has a given velocity and a position vector from the star. How do I find the magnitude of the gravity in the x, y and z directions.

$$positionvector = (x_0, y_0, z_0)$$
$$velocityvector = (v_x, v_y, v_z)$$
$$x = \frac{1}{2}*g_x*t^2 + v_x*t + x_0$$
$$y = \frac{1}{2}*g_y*t^2 + v_y*t + y_0$$
$$z = \frac{1}{2}*g_z*t^2 + v_z*t + z_0$$
$$r = \sqrt{x^2 + y^2 + z^2}$$
$$\theta = atan(\frac{y}{x})$$
$$\phi = acos(\frac{z}{r})$$
$$g = \sqrt{g_x^2 + g_y^2 + g_z^2} = -\frac{GM}{r^2}$$

Any hints on how to find $$(g_x, g_y, g_z)$$

Last edited: Aug 20, 2013
2. Aug 20, 2013

Staff: Mentor

That is not true, those formulas would require a constant acceleration. Acceleration in a gravitational field is not constant (even if that can be a good approximation in some cases).

Newton's law of gravity in its vector form gives that. Alternatively, use your g, and let it point from the planet to the central object.

3. Aug 20, 2013

Astrum

You're playing with the two-body problem, right? Why don't you use polar coordinates, it simplifies things greatly.

I'm not exactly sure what you're trying to do.

4. Aug 20, 2013

Philosophaie

I am looking for an object that is in freefall and its path towards a star from an initial velocity and position.

How do you formulate the acceleration of non-constant acceleration?

Last edited: Aug 20, 2013
5. Aug 20, 2013

Staff: Mentor

In the general case with more than 2 objects, there is no useful, closed formula to calculate the position for all times.
With just 2 objects, this is known as Kepler problem and has exact solutions.

6. Aug 22, 2013

Electric Red

7. Aug 23, 2013

tfr000

What is generally done is something like multiplying Gma / rab2 by (ra-rb) / rab, where ma is the mass of object a and rab is the distance between objects a and b.
(ra-rb) / rab forms "direction cosines" when you resolve the vectors with appropriate x, y, z coordinates.