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Question

Two stars each of one solar mass (=2×1030 kg) are approaching each other for a head on collision. When they are a distance 109 km, their speeds are negligible. What is the speed with which they collide ? The radius of each star is 104 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).

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Solution

Given:
The mass of a star is, M=2×1030 kg
The radius of each star is, R=104 km=107 m
Distance between two stars is,r=109 km=1012 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=GMMr ......(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
Ek=12Mv2+12Mv2=Mv2
and the total potential energy of the two stars system close to each other is:
Ep=GMM2R
Thus, the total energy of the two stars just before collision
Et=Mv2GMM2R ........(2)
Since the energy remains conserved, the initial and final energy will be same.
Mv2GMM2R=GMMr

v2=GM2RGMr
=6.67×1011×2×1030(12×10711012)
6.67×1012

v=2.58×106 m/s

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