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Two particles moving in circular orbits stopped suddenly. time to collide

  1. Aug 30, 2012 #1
    1. The problem statement, all variables and given/known data


    Two particles moving under the influence of their mutual gravitatonial force describe circular orbits about one another with period τ.if they are suddenly stopped in their orbits and allowed to gravitate toward each other. show that they will collide in a time τ/4√2.


    2. Relevant equations

    Since their orbits are circular. I use the following equation to find the period τ.
    μω2a=k/a2
    where a is the radius of the circular orbit and μ the reduced mass and k=Gm1m2
    ω=2[itex]\pi[/itex]/τ
    So i get τ2=μ/(ka34[itex]\pi[/itex]2)
    3. The attempt at a solution

    Now i apply conservation of energy.

    E=-k/a=1/2μ[itex]\dot{r}[/itex]2+l2/(2mr2)-k/r

    where l is the angular momentum

    from here i find [itex]\dot{r}[/itex]=dr/dt and the try to integrate properly from a to 0 to find the time and then somehow rearrange the expression so it is a function of τ.

    But i've checked the results and in turns out is wrong. In my book to calculate the time to collide they also apply cnservation of energy but like this:
    E= -k/a= 1/2μ[itex]\dot{x}[/itex]- k/x
    so the term with the angular momentum is not present.
    My question is why? when can you use each of these expressions for energy in a central force field? what is the difference? I can't figure it out i've had the same problem several times. I need some help.

    Thanks in advance.
     
  2. jcsd
  3. Aug 30, 2012 #2
    The kinetic energy of a particle is the product of its mass and the square of its velocity divided by two. That's all there is to it.
     
  4. Aug 30, 2012 #3
    Ok, but if i use polar coordinates the velocity can be decomposed in a radial and angular component. So then appears the term with l (angular momentum) which is constant in this central forces case. why can't i calculate the time to collide using that radial velocity?
     
  5. Aug 30, 2012 #4
    The radial velocity in circular motion is identically zero.

    I don't see why you need polar coordinates here. All you really need to figure out is the radius of the initial circular orbit, then, when the motion is stopped, that radius becomes the length along which the two particles accelerate toward each other.
     
  6. Aug 30, 2012 #5

    Sorry, perhaps i didn't explain myself properly. The radius of the circular orbit is known. When the motion is stopped why don't you need polar coordinates. I mean will the particles go in straight line toward each other. or will the reduced mass particle describe a straight line toward the center of force?
     
  7. Aug 30, 2012 #6
    i understand now. Even using polar coordinates when the particles stop l=0 and since l is a conserved quantity the term l2/(2mr2) will vanish.
    This will also mean that r and v are parallel at all times, l=m(r x v)
    Thank you very much for your time and help
     
    Last edited: Aug 30, 2012
  8. Aug 30, 2012 #7
    Are those two formulations really different? It is a straight line motion either way.
     
  9. Aug 31, 2012 #8
    no, of course, it is the same, but i needed to know why the angular momentum term vanished and i needed kind of a mathematical proof. that's all. i just understood while asking you.
    thank you
     
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