Question

Two stars each of one solar mass $(=2\times {10}^{30}kg)$ are approaching each other for a head on collision. When they are a distance ${10}^{9}km$, their speeds are negligible. What is the speed with which they collide ? The radius of each star is ${10}^{4}km$. Assume the stars to remain undistorted until they collide. (Use the known value of G).

Answer 1

Given:

The mass of a star is, $M=2\times {10}^{30}kg$
The radius of each star is, $R={10}^{4}km={10}^{7}m$
Distance between two stars is,$r={10}^{9}km={10}^{12}m$

As the stars are separated by a distance, initially the stars will have gravitational potential energy.

Total energy of the system of two stars is given by:

$E=-\frac{GMM}{r}$ ......(1)

Now as the stars move towards each other they gain the kinetic energy.

Let velocity of the stars when they are about to collide be $v$ and at collision the distance between the centers of the stars $=2R$

The total kinetic energy of the stars system

$E_k=\frac{1}{2}M{v}^2+\frac{1}{2}M{v}^2=M{v}^2$

and the total potential energy of the two stars system close to each other is:

$E_p=-\frac{GMM}{2R}$

Thus, the total energy of the two stars just before collision

$E_t=M{v}^2-\frac{GMM}{2R}$ ........(2)

Since the energy remains conserved, the initial and final energy will be same.
$M{v}^2-\frac{GMM}{2R}=-\frac{GMM}{r}$

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

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

$\approx 6.67\times {10}^{12}$

$\therefore v=2.58\times {10}^{6}m/s$