Position as a function of time throught a varying gravitational field

In summary, the conversation discusses the equations x=1/2at2 and a=gm/r2 and their combination to find the position x given the acceleration from gravity. The solution sought is a Kepler orbit and a relationship between fall time and distance can be found in [1] and [2]. The conversation concludes with the thanks for the helpful information.
  • #1
JJfortherear
14
0
I've been told this is a differential equation, but I have no knowledge of the subject, so I'm hoping it is and that an answer can be found in this forum. The two equations x=1/2at2 and a=gm/r2, when r is replaced with (r0-x), with r0 being the initial radius from the mass from which the gravitational field originates, and x being the distance traveled since t=0, due to the acceleration of the field. Combining these equations gives

x=1/2(gm/(r0-x)2)t2
I don't know how to solve that equation for x, and I'm hoping this is the place to look for help. Thanks.
 
Physics news on Phys.org
  • #2
The equation x = 1/2at2 is the position assuming constant acceleration a, so it does not make much sense to insert a variable acceleration, like from gravity, into it.

The solution you are looking for is a Kepler orbit [1], and if you have the degenerate case of purely radial motion then the solution is a so-called rectilinear Kepler orbit where the eccentricity is 1. For the special case where the initial speed is zero you can also find a relationship between fall time and distance in [2].

[1] http://en.wikipedia.org/wiki/Kepler_orbit
[2] http://en.wikipedia.org/wiki/Free_fall#Inverse-square_law_gravitational_field
 
  • #3
awesome, exactly what I was looking for. Thanks a lot.
 

1. What is the relationship between position and time in a varying gravitational field?

In a varying gravitational field, the position of an object changes over time due to the changing strength of the gravitational force acting on it. This can be described by the equation: x = x0 + v0t + 1/2at2, where x is the final position, x0 is the initial position, v0 is the initial velocity, t is time, and a is the acceleration due to gravity.

2. How does the strength of the gravitational field affect the position of an object over time?

The strength of the gravitational field directly affects the acceleration of an object, which in turn affects its position over time. The stronger the gravitational field, the greater the acceleration and the faster the object will change its position.

3. Can the position of an object in a varying gravitational field be predicted?

Yes, the position of an object in a varying gravitational field can be predicted using mathematical equations and principles such as Newton's laws of motion. By knowing the initial position, velocity, and acceleration of the object, its position at any given time can be determined.

4. How is time affected by a varying gravitational field?

In a varying gravitational field, time is also affected by the changing acceleration of an object. This is because time is a fundamental aspect of measuring the change in an object's position over time. As the acceleration changes, the time it takes for an object to reach a certain position also changes.

5. How does the position of an object in a varying gravitational field differ from that in a constant gravitational field?

In a varying gravitational field, the position of an object will change over time due to the changing acceleration. In a constant gravitational field, the position of an object will remain constant as long as it is not acted upon by any other forces. The equation for the position of an object in a constant gravitational field is: x = x0 + v0t - 1/2gt2, where g is the constant acceleration due to gravity.

Similar threads

A Is There a Solution for a Particle Falling into a Gravitational Well?
    • Differential Equations
Replies
3
Views
1K
The gravitating of a small mass towards a big mass
    • Introductory Physics Homework Help
Replies
3
Views
851
I Time Dilation in Gravitational Fields vs Acceleration on Earth
    • Special and General Relativity
Replies
7
Views
1K
I Explanation of all "its linearly independent derivatives"
    • Differential Equations
Replies
3
Views
1K
I On vibrating string and differentiating infinite sum
    • Differential Equations
Replies
7
Views
377
I Connection b/w curvature and gravitational field-removing coordinate transformations
    • Special and General Relativity
2
Replies
40
Views
2K
Gravitational equipotential multiple choice problem
    • Introductory Physics Homework Help
Replies
4
Views
750
A How to solve simple 2D space-time PDE numerically
    • Differential Equations
Replies
5
Views
2K
B Solving Kepler's 1st law as a function of time
    • Differential Equations
Replies
15
Views
2K
A Static Gravitational Field: Why Must ##g_{m0} = 0##?
    • Special and General Relativity
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top