Two neutron stars are separated by a distance of `1.0 xx 10^11` m. They each have a mass of `1.0xx10^30` kg and a radius of `2.0 xx 10^5` m. They are initially at rest with respect to each other. As measured from that rest frame, how fast are they moving when (a) their separation has decreased to one-half its initial value and (b) they are about to collide ?

Explanation

To solve the problem, we will use the principle of conservation of mechanical energy, which states that the total mechanical energy (kinetic + potential) in a closed system remains constant if only conservative forces are acting.

### Given:
- Mass of each neutron star, \( m = 1.0 \times 10^{30} \) kg
- Initial separation, \( r_{\text{initial}} = 1.0 \times 10^{11} \) m
- Radius of each neutron star, \( R = 2.0 \times 10^{5} \) m
- Gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)

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