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## 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.

μω

^{2}a=k/a

^{2}

where a is the radius of the circular orbit and μ the reduced mass and k=Gm

_{1}m

_{2}

ω=2[itex]\pi[/itex]/τ

So i get τ

^{2}=μ/(ka

^{3}4[itex]\pi[/itex]

^{2})

## The Attempt at a Solution

Now i apply conservation of energy.

E=-k/a=1/2μ[itex]\dot{r}[/itex]

^{2}+l

^{2}/(2mr

^{2})-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.