Gravitational Potential Energy question

[neo982]

Homework Statement
A binary star system has two stars, each with the same mass as our sun, separated by 1.00×10^12 m. 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?

The attempt at a solution

I know that this is supposed to use the K_f +U_f = K_i + U_i formula because energy is conserved. But the hard part to this problem is the reasoning behind it. I thing that the final potential energy (U_F) should be zero there fore giving me the eq. K_f = K_i + U_i...but then I start to wonder if the K_i should be zero also because it says the velocity is essentially at zero. When trying some of these ideas I end up with a radius (for the potential energy formula) -Gmm/r also I am not sure how to factor in the mass of both the planets and the radius between them, I initially thought that I should treat them as one big planet with total mass. I am close, but just need help with the conceptual part of this problem.

 

[rootX]

I think that the final potential energy (U_F) should be zero there fore giving me the eq.

Nopes, I don't think so
@ the midpoint it is r distance away from both stars

Therefore U_f = 2*(U with one star)

And using the formula GMm/r, you can find U_i (r is inf)

and K_f is just change in U


Edit: I think you are right for most of the part.. and in the end that should leave you with this equation:
K_f+U_f = 0

 

[neo982]

Thanks I got it.. K_f + U_f = 0 ..therefore v_f = sqrt([2*G(2*mass of planet)/r]) where radius is 1/2 of the value the question gives..which is the diameter...which is 32,586 m/s

Thanks.