Velocity of a satellite in an elliptical orbit

In summary, the conversation is about finding the velocities in an elliptical orbit using the equations for circular orbits and conservation of momentum. The person is struggling to find the necessary information to solve for the velocities and is looking for another equation that involves both Va and Vb. Another person suggests using conservation of angular momentum and potential energy to find a relationship between the velocities at the two points. The first person then discovers another equation that can be used to solve for the velocities at A and B. They also consider using conservation of energy as an alternative method.
  • #1
ehilge
163
0

Homework Statement


see attachment #12.106

Homework Equations


V=R[tex]\sqrt{}(g/r)[/tex] (for a circular orbit)
where R is the radius of the Earth and r is the radius of the orbit from the center of the earth

conservation of momentum for elliptical orbits:
Vara=Vbrb

The Attempt at a Solution


The first thing I did was find the velocity of the satellite while it is still in a circular orbit and came up with 1.46x108 m/s. Now this is fine and dandy but I don't see where there is enough information to get the velocities in the elliptical orbit since the only equation I have for an elliptical orbit is listed above and I don't have Va or Vb. I tried to pretend that the object also went into a circular orbit at B by another rocket boost. Hoping that I might be able to solve for something but to no avail. So I guess my question really is, what is another equation that also has Va and Vb so I can solve simultaneously, or is there a way to eliminate one of the variables that I don't see?
Thanks!
 

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  • #2
ehilge said:

Homework Statement


see attachment #12.106

Homework Equations


V=R[tex]\sqrt{}(g/r)[/tex] (for a circular orbit)
where R is the radius of the Earth and r is the radius of the orbit from the center of the earth

conservation of momentum for elliptical orbits:
Vara=Vbrb

The Attempt at a Solution


The first thing I did was find the velocity of the satellite while it is still in a circular orbit and came up with 1.46x108 m/s. Now this is fine and dandy but I don't see where there is enough information to get the velocities in the elliptical orbit since the only equation I have for an elliptical orbit is listed above and I don't have Va or Vb. I tried to pretend that the object also went into a circular orbit at B by another rocket boost. Hoping that I might be able to solve for something but to no avail. So I guess my question really is, what is another equation that also has Va and Vb so I can solve simultaneously, or is there a way to eliminate one of the variables that I don't see?
Thanks!
Angular momentum is conserved. So, [itex]m\omega r^2 = constant[/itex] or as you have noted, Vara=Vbrb.

You have Va and ra. What you need to find is rb - the effective radius at the farthest point. Would that not just be the distance from the centre of the earth? Be careful about translating radius from altitude.

AM
 
Last edited:
  • #3
ehilge said:
conservation of momentum for elliptical orbits:
Vara=Vbrb

This in general is not true for elliptical orbits. Conservation of momentum says [itex]\mathbf r \times \mathbf v[/itex] is constant. This is the vector cross product; the radial component of velocity is not involved in this expression. For elliptical orbits, the radial component of velocity is zero at two points: apogee and perigee. Thus [itex]r_av_a = r_p v_p[/itex] is valid.
 
  • #4
D H said:
This in general is not true for elliptical orbits. Conservation of momentum says [itex]\mathbf r \times \mathbf v[/itex] is constant. This is the vector cross product; the radial component of velocity is not involved in this expression. For elliptical orbits, the radial component of velocity is zero at two points: apogee and perigee. Thus [itex]r_av_a = r_p v_p[/itex] is valid.
Yes. The angular momentum is the tangential component of velocity divided by r. The confusion is avoided if one uses [itex]m\omega r^2 [/itex] for angular momentum.

AM
 
  • #5
Andrew Mason said:
Angular momentum is conserved. So, [itex]m\omega r^2 = constant[/itex] or as you have noted, Vara=Vbrb.

You have Va and ra. What you need to find is rb - the effective radius at the farthest point. Would that not just be the distance from the centre of the earth? Be careful about translating radius from altitude.


AM

first off, I realize that the radius at the furthest point is the altitude above the Earth + radius of the earth, and I have factored this into my calculations.

Secondly, I do not have Va of the object in the elliptical orbit, only in the original circular orbit. And I need to find the increase in speed the object makes at A while in a circular orbit to project it into an elliptical orbit. This is the part that I don't understand because I have two unknowns with the one equation.

D H said:
This in general is not true for elliptical orbits. Conservation of momentum says [itex]\mathbf r \times \mathbf v[/itex] is constant. This is the vector cross product; the radial component of velocity is not involved in this expression. For elliptical orbits, the radial component of velocity is zero at two points: apogee and perigee. Thus [itex]r_av_a = r_p v_p[/itex] is valid.

ok, fair enough and good to know. However, the points I am considering in the problem are at apogee and perigee, so the original equation I had is valid, right? And I still have the problem of having one equation with two unknowns. Any ideas on how to resolve that?
Thanks for your hep!
 
  • #6
Well there's always the conservation laws. You've already used conservation of angular momentum. What else is conserved in an orbit?

BTW, beware that the question used altitudes, not distances from the center of the Earth. You might find the latter to be a better choice. For example, in [itex]r_av_a = r_p v_p[/itex].
 
  • #7
ehilge said:
And I still have the problem of having one equation with two unknowns. Any ideas on how to resolve that?
What is the change in potential energy of the satellite in moving between A and B? Does that help you determine the relationship between speeds at A and B?

AM
 
  • #8
I managed to find another equation that I can use

1/rapogee+1/rperigee=2GM/h2 = 2(gR2)/(raVa)2 = 2(gR2)/(rbVb)2

I used this to find the velocity at A, and then calculated everything else from there using the previous equations.

I suppose I could also have used conservation of energy though

KEA + GPEA = KEB + GPEB

.5mvA2 + mgha = .5mvB2 + mghB

mass cancels out and I have another equation with va and vb

Thanks for your help!
 
  • #9
ehilge said:
I managed to find another equation that I can use

1/rapogee+1/rperigee=2GM/h2 = 2(gR2)/(raVa)2 = 2(gR2)/(rbVb)2

!
You have to explain where this formula comes from. Consider:

[tex]\Delta KE + \Delta PE = 0[/tex]

[tex]1/2(mv_A^2 - mv_B^2) + (-GMm/r_A + GMm/r_B) = 0[/tex]

[tex]v_A^2 - v_B^2 = 2(GM/r_A - GM/r_B)[/tex]

From the conservation of angular momentum:

[tex]v_Ar_A = v_Br_B[/tex]

so you can solve for vA

AM
 

1. What is the formula for calculating the velocity of a satellite in an elliptical orbit?

The formula for calculating the velocity of a satellite in an elliptical orbit is v = √(GM(2/r - 1/a)), where v is the velocity, G is the gravitational constant, M is the mass of the central body, r is the distance between the satellite and the central body, and a is the semi-major axis of the satellite's orbit.

2. How does the velocity of a satellite in an elliptical orbit vary throughout its orbit?

The velocity of a satellite in an elliptical orbit is not constant throughout its orbit. It is fastest at the perigee (closest point to the central body) and slowest at the apogee (farthest point from the central body). This is due to the varying distance between the satellite and the central body, as well as the changing strength of the gravitational force.

3. What is the difference between the velocity of a satellite in a circular orbit and an elliptical orbit?

In a circular orbit, the velocity remains constant throughout the orbit, while in an elliptical orbit, the velocity varies. Additionally, the velocity of a satellite in a circular orbit is directly proportional to the radius of the orbit, while in an elliptical orbit, it is dependent on both the distance from the central body and the semi-major axis of the orbit.

4. How does the velocity of a satellite in an elliptical orbit affect its orbital period?

The velocity of a satellite in an elliptical orbit is directly related to its orbital period. As the velocity increases, the orbital period decreases and vice versa. This is because the velocity and the distance from the central body are inversely proportional in the formula for calculating the orbital period.

5. Can the velocity of a satellite in an elliptical orbit be changed?

Yes, the velocity of a satellite in an elliptical orbit can be changed. This can be done through various methods such as using thrusters or gravity assists from other celestial bodies. However, any changes in velocity will also affect the satellite's orbit and may require careful planning and calculations.

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