Gravity in the x, y and z directions

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In summary, the conversation discusses finding the magnitude of gravity in the x, y, and z directions for an object orbiting a star. The formulas provided are not entirely accurate as they assume constant acceleration, but using Newton's law of gravity in vector form or using polar coordinates can help find the acceleration. This problem is known as the two-body problem and has exact solutions. Generally, the formula involves multiplying the product of the masses and the inverse square of the distance between the objects by the direction cosines.
  • #1
Philosophaie
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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.

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

Any hints on how to find [tex](g_x, g_y, g_z)[/tex]
 
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  • #2
Philosophaie said:
[tex]x = \frac{1}{2}*g_x*t^2 + v_x*t + x_0[/tex]
[tex]y = \frac{1}{2}*g_y*t^2 + v_y*t + y_0[/tex]
[tex]z = \frac{1}{2}*g_z*t^2 + v_z*t + z_0[/tex]
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).

Any hints on how to find [tex](g_x, g_y, g_z)[/tex]
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
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
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?
 
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  • #5
How do you formulate the acceleration of non-constant acceleration?
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.
 
  • #7
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.
 

What is gravity in the x direction?

Gravity in the x direction refers to the component of gravitational force acting on an object in the horizontal direction (left to right or right to left). This force is dependent on the mass of the object and the gravitational constant.

How does gravity in the y direction affect objects?

Gravity in the y direction is responsible for the vertical motion of objects. It determines the rate at which objects fall towards the earth and is also dependent on the mass of the object and gravitational constant.

What is the difference between gravity in the x and y directions?

The main difference between gravity in the x and y directions is the direction in which the force is acting. Gravity in the x direction acts horizontally while gravity in the y direction acts vertically.

How is gravity in the z direction related to the other two directions?

Gravity in the z direction is perpendicular to both the x and y directions and is responsible for the depth of an object's motion. It works in conjunction with gravity in the x and y directions to determine the overall motion of an object.

Can the direction of gravity be changed?

No, the direction of gravity cannot be changed. It always acts towards the center of the earth, and the direction of an object's motion is dependent on the direction of the gravitational force acting on it.

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