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Gravitation and its effect on elliptical orbits

  1. May 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Comets travel around the sun in elliptical orbits with large eccentricities. Suppose the comet has an initial speed of 1.17*10^4 m/s when at a distance of 4.9*10^11 m from the center of the sun, what is its speed when at a distance of 5.9*10^10 m? Give your in m/s in scientific notation to three significant digits. (Note: Use appendix F for the necessary data.)

    2. Relevant equations

    Don't know 'em. All I know is finding velocity in a circular orbit:

    v = sqrt((Gravitational constant * mass of the object being orbited around) / radius from the center of the orbited object)

    3. The attempt at a solution

    v = sqrt((6.67E-11 * 1.99E30) / 5.9E10) = 4.74E4

    The correct answer is: 6.40E4...so since I'm in the same order of magnitude I assume I'm close.
     
  2. jcsd
  3. May 7, 2009 #2

    mgb_phys

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    Science Advisor
    Homework Helper

    Orbits sweep out equal areas in equal times
    Consider the area of a triangle swept by the comet in 1 sec (eg some time short enough that you can make the path a straight line) when it's far and close to the sun.

    If the area is the same and you know the height you cna work out how long the base is and so the speed.
     
  4. May 7, 2009 #3
    O.K...so since the times and areas are equal, we can safely set both sides to equal the area of a triangle? i.e.

    .5 * R1 * V1 = .5 * R2 * V2
    thus,
    V2 = V1 * R1 / R2

    But then I get 9.72E4.
     
  5. May 7, 2009 #4
    There was no mention of
    perihelion or aphelion.

    Conservation of energy is what is expected

    David
     
    Last edited: May 7, 2009
  6. May 7, 2009 #5
    Alright, then here is what I got from davieddy's help:

    K + U = K + U

    MC = mass of the comet

    MS = mass of the sun

    G = gravitational constant

    So...here is my equation, followed by the steps to make it solve for V2:

    .5 * MC * V1 ^ 2 - G * MS * MC / D1 = .5 * MC * V2 ^ 2 - G * MS * MC / D2
    <=>
    .5 * V1 ^ 2 - G * MS / D1 = .5 * V2 ^ 2 - G * MS * D2
    <=>
    ...bunch of algebra...
    <=>
    V2 = (V1 ^ 2 + .5 * G * MS (D2 ^ -1 - D1 ^ -1)) ^ .5

    which leaves me with 3.36E4. Still in the right ballpark, but not quite there.
     
  7. May 7, 2009 #6
    Try
    V2 = (V1 ^ 2 + 2 * G * MS (D2 ^ -1 - D1 ^ -1)) ^ .5
     
  8. May 7, 2009 #7
    D'oh! Tyvm.
     
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