Gravitational Potential Energy

[iamazombie911]

Homework Statement

Two neutron stars are separated by a distance of 4.80 E 10 m. They each have a mass of 3.60 E 30 kg and a radius of 1.30 E 5 m. If they are initially at rest...
How fast is each star moving when their separation has decreased to half its initial value?
How fast is each star moving just before they collide?

Homework Equations

(G = Newton's Gravitational Constant (around 6.67*10^-11)[/B]
GPE = -G(m1)(m2)/r
Work = GravitationalForce*Distance = GPE
GravitationalForce = (G)(M1)(M2)/R^2

The Attempt at a Solution

Both forces from each planet are acting on each other, so I put the total gravitational force as double the equation. Multiplied by the given distance to get work/GPE, and then took that and subtracted the same equation, except with the distance halved. Don't really know where to go from here.

 

[haruspex]

First, you cannot add the two forces together like that. Adding those two forces produces zero, since they are just action and reaction of the same interaction.
Secondly, you cannot take the force to be constant over the distance. It will increase as they get closer.
For this question, it is easier to forget about forces and just think about energy.

 

[Kaura]

First, you cannot add the two forces together like that. Adding those two forces produces zero, since they are just action and reaction of the same interaction.
Secondly, you cannot take the force to be constant over the distance. It will increase as they get closer.
For this question, it is easier to forget about forces and just think about energy.

Remember that gravitational potential energy can be transferred into other energy types and that total energy at a given point will always be preserved and what haruspex said is true as I believe this problem is most easily solve by only taking the energy and change of energy type of the two bodies into account