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Derive equation of trajectory of a body around a fixed body attracted by gravity

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data

    There is a fixed spherical body of mass M whose center is to be taken as origin. Another body of mass m whose initial position vector [itex]\vec{r}[/itex] is given. This body is projected with initial velocity [itex]\vec{v}[/itex]. Find the equation of trajectory of body with mass m around the body with mass M.

    2. Relevant equations

    Will the trajectory be an ellipse, just like the orbit of earth around the sun?

    3. The attempt at a solution

    I tried solving the position using Newton's Law of Gravity. I also tried using the formula a = v dv/dx and integrating but was unable to reach a solution.

    Any help is greatly appreciated...
     
  2. jcsd
  3. Apr 9, 2012 #2

    mfb

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    2016 Award

    Staff: Mentor

    Your problem is known as Kepler problem.

    It is possible to derive the trajectory with Newton's Law of gravity, but this is an ugly calculation, at least ~2 pages long, involving polar coordinates, some substitutions and messy integrals.

    Depending on the velocity, the radius and the masses M and m, the trajectory can be:
    - an ellipse
    - a parabola
    - a hyperpola
    which are all conic sections
     
  4. Apr 10, 2012 #3
    thank you mfb
    but i am willing to go through the messy mathematics. So can u please help me by giving me a link to where this Kepler problem has been solved.
     
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