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Finding the speed of the comet

  1. Dec 11, 2016 #1
    1. The problem statement, all variables and given/known data

    Comets travel around the sun in elliptical orbits with large eccentricities. If a comet has speed 2.2×10^4 m/s when at a distance of 2.6×10^11 m from the center of the sun, what is its speed when at a distance of 4.2×10^10 m
    2. Relevant equations
    L = rp = r(mv)

    3. The attempt at a solution
    I thought angular momentum is what I had to use for this one and thought of the two distances as perigee and apogee. I used conservation of momentum to attempt to solve the speed:
    rmv = rmv
    the m's cancel out
    so im left with r1v1 = r2v2
    > 2.6 * 10^11m(2.210^4m/s)=4.2*10^10m(v2)
    and I solved for the speed like that and got 136190. But that is wrong.
     
  2. jcsd
  3. Dec 11, 2016 #2

    gneill

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    Staff: Mentor

    Angular momentum would be a good approach if the two positions in question were aphelion and perihelion. Then the velocities would be perpendicular to the radii and you could apply conservation of angular momentum in a simple scalar form. But this problem does not say that the given positions are aphelion or perihelion, so there's no knowing what the angular relationship will be between the position and velocity vectors.

    What other conservation law might you appeal to instead?
     
  4. Dec 11, 2016 #3
    Could conservation of mechanical energy work?
    1/2mv2=GmM/r
     
  5. Dec 11, 2016 #4

    gneill

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    Yes...
    ...But not that way. The KE won't be equal to the PE (except for very specific circumstances that don't arise here).

    What's the expression for the total mechanical energy of a body in orbit? Note that you can ignore the mass of the object itself and use what's known as the Specific Mechanical Energy. That's the energy per kg for the body in orbit.
     
  6. Dec 11, 2016 #5
    Would it be E=-1/2(GmM/r)
     
  7. Dec 11, 2016 #6

    gneill

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    That only accounts for the gravitational potential energy (actually half of the GPE). What's missing?
     
  8. Dec 11, 2016 #7
    woops.
    So is it
    E= 1/2mv^2 - GmM/r ?
     
  9. Dec 11, 2016 #8

    gneill

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    Yes, that's the total mechanical energy. As I mentioned previously, you can drop the "m" from the terms and it becomes the Specific Mechanical Energy which works just as well for these problems. You'll find that the "m" would eventually cancel out anyways if keep it.
     
  10. Dec 11, 2016 #9
    Okay so then all we have to worry about is gnna be
    1/2v^2 - GM/r ?
     
  11. Dec 11, 2016 #10

    gneill

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    Yup. That will be the specific mechanical energy, which is a constant for the entire orbit.
     
  12. Dec 11, 2016 #11
    So how do i manipulate this to find the missing velocity given two radii and one velocity?
     
  13. Dec 11, 2016 #12

    gneill

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    The value that the equation delivers is a constant of the orbit. It applies everywhere along the trajectory.

    Think in terms of conservation of energy problems that you've done for objects near the Earth's surface, where you equate initial and final energy sums.
     
  14. Dec 11, 2016 #13
    Oh okay so it's going to be like 1/2v^2 - GM/r = 1/2v^2 - GM/r?
     
  15. Dec 11, 2016 #14

    gneill

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    Yes, that's the idea.
     
  16. Dec 11, 2016 #15
    But I do not have the M, so do i have to solve for that first by using another equation?
     
  17. Dec 11, 2016 #16

    gneill

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    Often the gravitational parameter μ = GM is a given value. Your course materials or textbook likely provide a value for it. If not μ then the mass of the Sun should be given.
     
  18. Dec 11, 2016 #17
    Thank you so much for the help. I got the right answer finally! :)
     
  19. Dec 11, 2016 #18

    gneill

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    I'm glad I could help!
     
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