Stars Accelerating towards each other

[spaderdabomb]

Homework Statement

Two stars move in circular orbits about one another with period T due to
their mutual gravitational attraction. If the objects are suddenly stopped, show that they
will then collide with one another after a time T /(4√2)

Homework Equations

T = ∫dt
dA/dt = (1/2)r²d∅/dt
F_g= -Gm₁m₂/r²

The Attempt at a Solution

So I believe I've determined the period in terms of radius. Simply by substituting what dt equals into the integral and solving. You get T = 2piμr²/4l, where l is the magnitude of the angular momentum, and μ is the reduced mass.

After this I'm a little stuck. I've tried integrating the acceleration given by the force on each star, but I get stuck with the differential equation dv = -Gm₁/(4r²). Maybe this is the totally wrong way to approach it. In that equation r is not a constant, therefore you can't just integrate it. If doing it this way, you'd have to replace r with a constant multiplied by something having to do with time, but I don't know what that would be.

And also, regular kinematic equations can't be used of course since this is not constant acceleration. Thanks in advance!

 

[haruspex]

Conservation of momentum gives you that the mass centre of the system does not move, so measuring distances from there allows you to concentrate on how long one star will take to reach that point.
How about cheating and using conservation of energy to avoid one level of integration?

 

[spaderdabomb]

Hmm ok...I've tried using energy and momentum and it gave me a little progress...I can only get velocity in terms of distance, or vice-versa...so I'm still having trouble getting distance in terms of time =(. Any suggestions?

 

[haruspex]

Any suggestions?

Yes: post what you have so far