Two neutron stars are separated by a distance of 10^14 m. They each have a mass of 10^30 kg and a radius of 10^5 m. They are initiall at rest with respect to each other.

Question

How fast are they moving when their seperation has decreased to one-half its initial value?
How fast are they moving before they collide?

Answer 1

Let R be the initial separation and r be thr radius of each.

The initial potential energy is
PE1 = -GM^2/R

When the separation is R/2, the potential energy is
PE2 = -2GM^2/R

The reduction in potential energy, GM/R^2, is converted to kinetic energy equal to
2 *(1/2) M V^2, where V is the velocity of each towards the center of mass. Solve for V

V^2 = GM/R
V= sqrt(GM/R)

G is the universal constant of gravity.

When they collide, the potential energy reduction is
GM^2[(1/r) - (1/R)]

Set that equal to 2(1/2) M V'^2 for the velocity of each, V'.

If it is close to the speed of light, a relativistic equation should have been used.