Mass of each star, M = 2 × ${10}^{30}$ kg

Radius of each star, R =${10}^4$ km =${10}^7$ m

Distance between the stars, r = ${10}^9$ km = ${10}^{12}\mathrm{m}$

For negligible speeds, v = 0 total energy of two stars separated at distance r

$=\frac{-\mathrm{GMM}}{\mathrm{r}}+\frac{1}{2}{\mathrm{mv}}^2$
$=\frac{-\mathrm{GMM}}{\mathrm{r}}+0 ........\left(\mathrm{i}\right)$

Now, consider the case when the stars are about to collide:

Velocity of the stars = v

Distance between the centers of the stars = 2R

Total kinetic energy of both stars $=\frac{1}{2}{\mathrm{Mv}}^2+\frac{1}{2}{\mathrm{Mv}}^2={\mathrm{Mv}}^2$

Total potential energy of both stars$=\frac{-\mathrm{GMM}}{2\mathrm{R}}$

Total energy of the two stars = ${\mathrm{Mv}}^2-\frac{\mathrm{GMM}}{2\mathrm{R}} .......\left(\mathrm{ii}\right)$

Applying the law of conservation of energy,

$M{v}^{2 }-\frac{GMM}{2R}=-\frac{GMM}{r}$

${\mathrm{v}}^2=\frac{-\mathrm{GM}}{\mathrm{r}}+\frac{\mathrm{GM}}{2\mathrm{R}}=\mathrm{GM}\left(-\frac{1}{\mathrm{r}}+\frac{1}{2\mathrm{R}}\right)$
$=6.67\times {10}^{-11}\times 2\times {10}^{30}\left[-\frac{1}{{10}^{12}}+\frac{1}{2\times {10}^7}\right]$
$=13.34\times {10}^{19}[-{10}^{12}+5\times {10}^{-8}]$
$~13.34\times {10}^{19}\times 5\times {10}^{-8}$
$~6.67\times {10}^{12}$
$\mathrm{v}=\sqrt{6.67\times {10}^{12}}=2.58\times {10}^6\mathrm{m}/\mathrm{s}$