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Orbit Intersection Question (classical mechanics)

  1. Jan 20, 2007 #1
    1. The problem statement, all variables and given/known data
    A comet is going in a parabolic orbit lying in the plane of Earth's Orbit. Regarding Earth's orbit as circular of radius "a," show that the points where the comet intersects Earth's orbit are given by:

    cos(theta)= -1 + (2*p)/a where p is the perihelion distance of the comet defined at (theta)=0


    2. Relevant equations
    Circular orbit: eccenticinty = 0
    Parabolic orbit, Energy = 0, eccentincity = 1

    Differential equation of an orbit:

    d^2(u)/d(theta)^2 + u = -1/(m*l^2*u^2)*f(u^-1), where f is the function of the central force, u= 1/r, where r is the radius, and l is the angular momentum per mass.

    Another representation of the differential equation of an orbit (using energy, and only for an inverse-squared central attractive force):

    (du/d(theta))^2 + u^2 = 2E/ml^2 + 2ku/(ml^2); conditions are the same, except k= GM.

    3. The attempt at a solution
    Ok my thought was the solve the differential equations, once I have the solutions set the equations equal to one another and show that the points are predicted by the equation given; however, I don't have enough information given to solve the diff. eqs.

    Any other ideas? I am quite stumped.
     
  2. jcsd
  3. Jan 20, 2007 #2

    D H

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    Stop using physics. This is a simple geometry problem: Find the intersection of a parabola and a circle. The only physical result that is needed is that the center of the circle and the focus of the parabola are the same point.
     
  4. Jan 20, 2007 #3

    Dick

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    You don't have to solve the equations of motion. The geometry of the orbits is already given to you. So it's just a geometry problem. Just write down the polar coordinate representations of the two curves and intersect them. It's REALLY easy.
     
  5. Jan 20, 2007 #4
    Thanks for the tips; and yes you were right it was so easy to just think of it in geometric terms.
     
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