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

[cler]

Homework Statement

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.

Homework 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\pi/τ
So i get τ²=μ/(ka³4\pi²)

The Attempt at a Solution

Now i apply conservation of energy.

E=-k/a=1/2μ\dot{r}²+l²/(2mr²)-k/r

where l is the angular momentum

from here i find \dot{r}=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\dot{x}- 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.

 

[voko]

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.

 

[cler]

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.

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?

 

[voko]

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.

 

[cler]

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.

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?

 

[cler]

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

 

[voko]

Are those two formulations really different? It is a straight line motion either way.

 

[cler]

Are those two formulations really different? It is a straight line motion either way.

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