Gravitational Forces of binary star

[nitrostar]

I have this problem:

The question says: A binary star system has two stars, each with the same mass as our sun, separated by 1x10^12m. A comet is very far away and essentially at rest. Slowly but surely, gravity pulls the comet toward the stars. Suppose the comet travels along a straight line that passes through the midpoint between the two stars. What is the comet's speed at the midpoint?

I set up the system like this:

      c
    / | \
   /  |  \
  /   |   \
bs1---m---bs2

where c=comet
m = midpoint
bs1 and bs2 are the binary stars

We know Fxnet=0
And Fynet = 2*Fybs2
Fbs2=mMG/r^2
if we set up an angle @ between the commet and bs2 we can deduce that
sin@=5*10^11/r
so
r=5*10^11/sin@

Fbs2=mMG/(5*10^11/sin@)

Fybs2=Fbs2*cos@
Fybs2=mMG*sin@*cos@/(5*10^11)


Fynet=(2*mMG/(5*10^11))*int(sin@cos@d@,0,pi/2)
Fynet=m*2.655*10^8=ma
a=2.655*10^8

But now what??
Am i doing things right?

Thank you very much!
nitro

 

[andrevdh]

If one measures the distance of the comet from the midpoint between the two stars, $r$, up to the point where it currently is, then the resulting force from the two stars points from the comet towards this midpoint and is given by twice the component of one of these forces, say $F_S$, along $r$. This results in an equation for the resultant force in terms of $G,M,m,r\ and\ d$ where d is half the distance between the stars. Apply the work kinetic energy theorem - that is integrate this force over $r$ for r from infinity to zero. This gives you the kinetic energy when the comet reaches the midpoint between the two stars.

 

[nitrostar]

i solved it using potential/kinetic energy and the law of conservation of energy!

Thank you very much though!