Question

Two neutron stars are separated by a distance of 1.0 x 10¹¹ m. They each have a mass of 1.0 x 10²⁹ kg and a radius of 1.0 x 10⁵ 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?

Answer 1

Use the principle of conservation of energy. The initial potential energy is

Uo =-GM^2/ri
The initial kinetic energy is zero since the stars are at rest.
The final potential energy is
Uf = -2GM^2/ri
-GM^2/ri = -2GM^2/ri + Mu^2
u= root G*M/ri
= root 6.67*10^-11M^3/s^2kg(10^29)/10^11 m
= 8167 m/s

Now the final separation of the centers is rf = 2R = 2*10^5
-GM^/ri = -GM^/rf+ Mu^2
u = root GM (1/rf-1/ri)
  

root 6.67*10^-11M^3/s^2kg(10^29)(1/2*10^5 - 1/10^11)
= 5774940.12 m/s