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

 

[paranormal]

I would approach this problem by using the concept of conservation of energy. When the stars are in circular orbits, they have a certain amount of kinetic energy and gravitational potential energy. When they suddenly stop, their kinetic energy becomes zero, but their gravitational potential energy remains the same. This means that the total energy of the system remains constant.

We can use the equation for gravitational potential energy, U = -Gm1m2/r, to determine the distance between the two stars at any given time. Since the total energy remains constant, we can set the initial potential energy equal to the final potential energy and solve for the distance r.

U(initial) = U(final)
-Gm1m2/r(initial) = -Gm1m2/r(final)
r(final) = r(initial)

This tells us that the distance between the two stars remains constant after they stop moving. We can then use this distance and the equation for circular motion, v = √(GM/r), to determine the initial velocity of the stars before they stopped.

v = √(GM/r(initial))

Now, using the distance and initial velocity, we can calculate the time it takes for the stars to collide using the equation for circular motion, T = 2πr/v. This gives us the time T/(4√2) as stated in the homework statement.

In summary, by using the concept of conservation of energy, we can determine that the distance between the two stars remains constant after they stop moving. We can then use this distance and the equation for circular motion to determine the initial velocity of the stars. Finally, we can use the equation for circular motion again to calculate the time it takes for the stars to collide.