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Gravitational question ( calculus )

  1. May 10, 2012 #1
    1. The problem statement, all variables and given/known data
    2 point mass M1 and M2 . Their separation is r0. They are released at rest at t=0. Find the time of collision (T)of them due to gravitational attraction in terms of M1, M2 , G and r0.

    2. Relevant equations
    F=ma
    ∫a(dt)2 = s (a is acceleration, s is displacement)
    x0=(M2/(M1 + M2))r0 ( x is the position of center of mass respect to M1 )
    F=-GM1M2/r2

    3. The attempt at a solution
    I used ∫a(dt)2 = s and make it becomes dt=(v/a)ds. As they should collide at the center of mass, the displacement of M1 should be equals to x. Therefore,
    T=∫(v/a)dx
    sup x=(M2/(M1 + M2))r , let (M2/(M1 + M2)) = R
    x=Rr
    dx=Rdr
    T=R∫(v/a)dr (from r=r0 to r=0)
    By law of conservation of energy,
    (1/2)M1v2 = GM1M2(1/r - 1/r0)
    v=√(2GM2(1/r - 1/r0))
    a=GM2/r2
    ∴T=R∫√(2GM2(1/r - 1/r0)) / (GM2/r2) dr
    After some calculation, I got
    T=R√(2/(GM2r02))∫r2√(1/r - 1/r0)dr (from r=r0 to r=0)
    I found that integration is very difficult. I wonder whether I get the correct direction or having conceptional and calculating error. I just hope that any help can be given. Thanks a lot.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 12, 2012 #2
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