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Universal gravity - check my answer

  1. Mar 11, 2009 #1
    1. The problem statement, all variables and given/known data

    Mercury's orbital speed varies from 38.8km at aphelion to 59.0km at perihelion.

    If the planet is 6.60*10^10m from the Sun's center at aphelion, how far is it at perihelion?

    2. Relevant equations

    v = sqrt(GM/r) (speed,circular orbit)


    3. The attempt at a solution

    V_a = velocity of aphelion = 38.8km = 3.88 * 10^4m
    V_p = velocity of perihelion = 59.0km = 5.9 * 10^4m


    V_a = sqrt(GM/r)

    3.88 * 10^4m = sqrt(GM/r_a)
    and solving for M gives me (I think)

    M = (3.88 * 10^4m*r_a)^2 / G


    V_b = sqrt(GM/r_p) ; where W_b = 5.9 * 10^4m

    solving for r_p gives :

    r_p = (3.88 * 10^4m *r_a ) / 5.9 * 10^4m

    ; where r_a = 6.60*10^10m;

    and

    r_p = 4.60 * 10^10

    is this right?
     
  2. jcsd
  3. Mar 11, 2009 #2

    AEM

    User Avatar

    Since Mercury travels in an ellipse you shouldn't be using an equation for velocity in a circular orbit. What you want to do is to use the fact that total energy is conserved. That is,

    KE (Perihelion) + PE (perihelion) = KE (aphelion) + PE (aphelion)

    Once you express these energies in their proper form, you will see that you have only one unknown -- the distance at perihelion.
     
  4. Mar 11, 2009 #3
    Is this the right equation ;

    KE (Perihelion) + PE (perihelion) = KE (aphelion) + PE (aphelion)

    0 - GMm/ r_e = 1/2 mv^2 - GMm/r_a

    and solving for r_e I get (check it )

    r_e = GM / (1/2 v^2) - r_a

    and plug in the numbers now, if this is correct
     
    Last edited: Mar 11, 2009
  5. Mar 11, 2009 #4
    anyone?
     
  6. Mar 12, 2009 #5

    LowlyPion

    User Avatar
    Homework Helper

    How did you get that KE at perihelion was 0?
     
  7. Mar 12, 2009 #6
    ok wait here is what i think is correct,

    KE (Perihelion) + PE (perihelion) = KE (aphelion) + PE (aphelion)

    1/2*mv^2 - GMm/ r_e = 1/2 mv^2 - GMm/r_a

    = (check this below?)

    r_e = -GMm / (1/2mV_a^2 - GMm/r_a -1/2mV_p^2 )

    r_e = -GM / (.5V_a^2 - GM/r_a - .5V_p^2 )

    r_e = 4.598 * 10^10
     
  8. Mar 12, 2009 #7
    I guess that is correct but I have 1 question :

    KE (Perihelion) + PE (perihelion) = KE (aphelion) + PE (aphelion)

    it does not matter if

    the above formual is switch right, the perihelion on the right side and
    the aphelion on the left?
     
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