Answer
The speed of each neutron star is $~~1.8\times 10^7~m/s$
Work Step by Step
Let $R_1 = 1.0\times 10^{10}~m$
We can find the separation distance of the centers when they are about to collide:
$R_2 = (2)(1.0\times 10^5~m) = 2.0\times 10^5~m$
We can use conservation of energy to find the kinetic energy of the system when the separation distance is $\frac{R_0}{2}$:
$K_2+U_2 = K_1+U_1$
$K_2-\frac{GM^2}{R_2} = 0-\frac{GM^2}{R_1}$
$K_2 = \frac{GM^2}{R_2}-\frac{GM^2}{R_1}$
$K_2 = GM^2~(\frac{1}{R_2}-\frac{1}{R_1})$
Since both neutron stars have the same mass, each star has half of the kinetic energy of the system.
We can find the speed of each neutron star:
$K = \frac{GM^2}{2}~(\frac{1}{R_2}-\frac{1}{R_1})$
$\frac{1}{2}Mv^2 = \frac{GM^2}{2}~(\frac{1}{R_2}-\frac{1}{R_1})$
$v^2 = GM~(\frac{1}{R_2}-\frac{1}{R_1})$
$v = \sqrt{GM~(\frac{1}{R_2}-\frac{1}{R_1})}$
$v = \sqrt{(6.67\times 10^{-11}~N~m^2/kg^2)(1.0\times 10^{30}~kg)~(\frac{1}{2.0\times 10^5~m}-\frac{1}{1.0\times 10^{10}~m})}$
$v = 1.8\times 10^7~m/s$
The speed of each neutron star is $~~1.8\times 10^7~m/s$
