Colliding Binary Star (GPE Problem)

[macaholic]

I was looking over my old physics course problems, and I can't figure out how I'm doing this one wrong.

Homework Statement

Two identical stars, each having mass and radius M=2*10^29 kg and R = 7 *10^8 m are initially at rest in outer space. Their initial separation (between centers) is the same as the distance between our sun and the earth, D = 1.5*10^11 m. Their gravitational interaction causes the stars to be pulled toward one another. Find the speed of the stars just before they collide, i.e. when their centers are a distance 2R apart.

Homework Equations

GPE = \frac{- G m_1 m_2}{r}
KE = \frac{m v^2}{2}

The Attempt at a Solution

I tried just doing conservation of energy, i.e.

GPE + GPE = GPE + GPE + KE + KE, or more explicitly:
\frac{-G M^2}{D} + \frac{-G M^2}{D} = \frac{-G M^2}{2R} + \frac{-G M^2}{2R} + \frac{1}{2} M v^2 + \frac{1}{2} M v^2

However solving this does NOT get the right answer, which is 9.7*10^4 m/s.

Can anyone point out what I'm doing wrong? I can't find the flaw in my logic... Does it have to do with where I'm setting zero potential energy? I tried accounting for this by doing the problem another way:
\Delta GPE = \Delta KE
But that seems to be equivalent to what I did above.

 

[Dick]

I was looking over my old physics course problems, and I can't figure out how I'm doing this one wrong.

Homework Statement

Two identical stars, each having mass and radius M=2*10^29 kg and R = 7 *10^8 m are initially at rest in outer space. Their initial separation (between centers) is the same as the distance between our sun and the earth, D = 1.5*10^11 m. Their gravitational interaction causes the stars to be pulled toward one another. Find the speed of the stars just before they collide, i.e. when their centers are a distance 2R apart.

Homework Equations

GPE = \frac{- G m_1 m_2}{r}
KE = \frac{m v^2}{2}

The Attempt at a Solution

I tried just doing conservation of energy, i.e.

GPE + GPE = GPE + GPE + KE + KE, or more explicitly:
\frac{-G M^2}{D} + \frac{-G M^2}{D} = \frac{-G M^2}{2R} + \frac{-G M^2}{2R} + \frac{1}{2} M v^2 + \frac{1}{2} M v^2

However solving this does NOT get the right answer, which is 9.7*10^4 m/s.

Can anyone point out what I'm doing wrong? I can't find the flaw in my logic... Does it have to do with where I'm setting zero potential energy? I tried accounting for this by doing the problem another way:
\Delta GPE = \Delta KE
But that seems to be equivalent to what I did above.

You are overcounting the potential energy. The gravitational potential energy of two masses of mass m separated by a distance r is -G*m*m/r. It's not twice that. You are counting the same thing twice.

 

[macaholic]

*facepalm*. Thank you! I feel very silly now.