Find the Position of an Orbiting Spacecraft

[Bluskyz]

Homework Statement

Lets say that there is a small spacecraft (mass doesn't matter due to Earth's huge mass) orbiting Earth with a current distance from the center at 6,700,000 m. This whole problem is on a 2D plane with the spacecraft 's current position at (0,6700000). It's current velocity is at 8,000 m/s along the x-axis. I would like to know how to determine the spacecraft 's x and y coordinates at any point in time. I've made this problem only to satisfy my mathematical curiosity.

Homework Equations

Acceleration to Earth: F=GM/r^2
G= Gravitational Constant = 6.67384e-11 (m^3/kg*s^2)
Earth's Mass: 5.97e24

The Attempt at a Solution

I have tried multiple ways to get the solution and most of them ended up nowhere. So, please, can you explain how to get the answer? Thanks for your replies.

 

[berkeman]

Homework Statement

Lets say that there is a small spacecraft (mass doesn't matter due to Earth's huge mass) orbiting Earth with a current distance from the center at 6,700,000 m. This whole problem is on a 2D plane with the spacecraft 's current position at (0,6700000). It's current velocity is at 8,000 m/s along the x-axis. I would like to know how to determine the spacecraft 's x and y coordinates at any point in time. I've made this problem only to satisfy my mathematical curiosity.

Homework Equations

Acceleration to Earth: F=GM/r^2
G= Gravitational Constant = 6.67384e-11 (m^3/kg*s^2)
Earth's Mass: 5.97e24

The Attempt at a Solution

I have tried multiple ways to get the solution and most of them ended up nowhere. So, please, can you explain how to get the answer? Thanks for your replies.

Welcome to the PF.

You have correctly listed one of the Relevant Equations (gravitational force), but you are missing the equations for uniform circular motion (motion under the constant centripital acceleration of gravity).

Can you list those equations? They will relate the centriptal force to the angular velocity and other helpful things...

(If you aren't sure what they are, just Google the terms I used, or check wikipedia.org)

 

[HallsofIvy]

Homework Statement

Lets say that there is a small spacecraft (mass doesn't matter due to Earth's huge mass) orbiting Earth with a current distance from the center at 6,700,000 m. This whole problem is on a 2D plane with the spacecraft 's current position at (0,6700000). It's current velocity is at 8,000 m/s along the x-axis.

How are the axes set up relative to the earth? We need to know that before we can answer. If you set the axes up with the origin at the center of the Earth then the spacecraft is moving either directly toward or away from the earth. (And we don't know which because you did not give the direction of motion of the velocity.)

I would like to know how to determine the spacecraft 's x and y coordinates at any point in time. I've made this problem only to satisfy my mathematical curiosity.

Homework Equations

Acceleration to Earth: F=GM/r^2
G= Gravitational Constant = 6.67384e-11 (m^3/kg*s^2)
Earth's Mass: 5.97e24

The Attempt at a Solution

I have tried multiple ways to get the solution and most of them ended up nowhere. So, please, can you explain how to get the answer? Thanks for your replies.

Homework Statement

Yes, the acceleration, a (I wouldn't use "F" for acceleration) is equal to GM/r^2. Also a= dv/dt= d^2r/dt^2= GM/r^2. We can integrate that using a method called "quadrature". Since v= dr/dt, by the chain rule, dv/dt= (dv/dr)(dr/t)= v dv/dr so that equation becomes v dv/dr= GM/r^2, a separable first order differential equation. Separating gives vdv= (GM/r^2)dr. Integrating, (1/2)v^2= -(GM/r)+ C. That can be written as (1/2)v^2+ GM/r= C which, if you multiply through by m, the mass of the ship, is "conservation of energy". Since v= dr/dt, we can solve (1/2)v^2= -(GM/r)+ C for v- v= \sqrt{2(C- (GM/r)} and then integrate both sides of dx= \sqrt{2(C- GM/r)}.

That last equation will not be easy to integrate! It is, in fact, a simple example of type of integral called an "elliptic integral", so called precisely because it is involved in problems like this. And the more general problem, where the ship or other object is NOT moving directly on a line to or form the earth, typically gives elliptic orbits.

Homework Equations

The Attempt at a Solution

[/QUOTE]

 

[HallsofIvy]

Welcome to the PF.

You have correctly listed one of the Relevant Equations (gravitational force), but you are missing the equations for uniform circular motion (motion under the constant centripital acceleration of gravity).

Good point. I looked at the simple case- where the object is moving directly toward or away from the earth. "Uniform circular motion" is perhaps a bit more general. But general orbital motion is NOT "uniform circular motion". More common is elliptical motion where the speed depends on the distance from the earth.

Can you list those equations? They will relate the centriptal force to the angular velocity and other helpful things...

(If you aren't sure what they are, just Google the terms I used, or check wikipedia.org)

 

[Bluskyz]

The origin is centered on the center of Earth and the spacecraft is moving 8,000 m/s in the x direction. Initially, no motion on the y axis.

As for the uniform circular motion equations, don't you need to have it's rate of rotation around Earth ( the origin)? Like degrees per second

 

[Bluskyz]

Good point. I looked at the simple case- where the object is moving directly toward or away from the earth. "Uniform circular motion" is perhaps a bit more general. But general orbital motion is NOT "uniform circular motion". More common is elliptical motion where the speed depends on the distance from the earth.

Yes, I'm pretty sure it is elliptical motion, the spacecraft 's speed changes due to it's acceleration towards Earth given by GM/r^2.

 

[gneill]

For a non-circular orbit the 'prediction problem' is not trivial. There are several mathematical methods available, most of which which involve solving a transcendental function. Investigate "The Kepler Problem" and "The Kepler Equation".

If you're keen on this topic I suggest picking up the the book "Fundamentals of Astrodynamics" by Bate, Mueller, and White. It's very inexpensive and covers the topic well and with plenty of examples and problems.

 

[Bluskyz]

For a non-circular orbit the 'prediction problem' is not trivial. There are several mathematical methods available, most of which which involve solving a transcendental function. Investigate "The Kepler Problem" and "The Kepler Equation".

If you're keen on this topic I suggest picking up the the book "Fundamentals of Astrodynamics" by Bate, Mueller, and White. It's very inexpensive and covers the topic well and with plenty of examples and problems.

Thanks for your reply. I have seen that book referenced many times on this subject so I think I'm going to get it. :)