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Q: 8.20 Two stars each of one solar mass ( $\small =2 \times 10^3^0\hspace {1mm}kg$ ) are approaching each other for a head on collision. When they are a distance $\small 10^9\hspace{1mm}km$ , their speeds are negligible. What is the speed with which they collide? The radius of each star is $\small 10^4\hspace{1mm}km$ . Assume the stars to remain undistorted until they collide. (Use the known value of G).

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Total energy of stars is given by :

$E\ =\ \frac{-GMM}{r}\ +\ \frac{1}{2}mv^2$

or $=\ \frac{-GMM}{r}\ +\ 0$

or $=\ \frac{-GMM}{r}$

Now when starts are just to collide the distance between them is 2R.

The total kinetic energy of both the stars is :

$=\ \frac{1}{2}mv^2\ +\ \frac{1}{2}mv^2\ =\ mv^2$

And the total energy of both the stars is :

$=\ mv^2\ +\ \frac{-GMM}{2r}$

Using conservation of energy we get :

$mv^2\ +\ \frac{-GMM}{2r}\ =\ \frac{-GMM}{r}$

or $v^2\ =\ GM \left ( \frac{-1}{r}\ +\ \frac{1}{2R} \right )$

or $=\ 6.67\times 10^{-11}\times 2\times 10^{30} \left ( \frac{-1}{10^{12}}\ +\ \frac{1}{2\times 10^7} \right )$

or $=\ 6.67\times 10^{12}$

Thus the velocity is :

Posted by

Devendra Khairwa

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