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### Gravitation NCERT Solutions Class 11 Physics - Solved Exercise Question 8.20

Question 8.20:
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).
Solution:
Mass of each star, M = 2 × 1030 kg
Radius of each star, R = 104 km = 107 m
Distance between the stars, r = 109 km = 1012m
For negligible speeds, v = 0 total energy of two stars separated at distance r
= [ -GMM / r ] + (1/2)mv2
[ -GMM / r ] + 0  ....(i)
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 = (1/2) Mv2 + (1/2)Mv2 = Mv2
Total potential energy of both stars = -GMM / 2R
Total energy of the two stars = Mv2 - GMM / 2R    ....(ii)
Using the law of conservation of energy, we can write:
Mv2 - GMM / 2R =  -GMM / r
v2 = -GM / r + GM / 2R
= GM [ (-1/r) + (1/2R) ]
= 6.67 X 10-11 X 2 X 1030 [ (-1/1012 ) + (1 / 2X107) ]
~ 6.67 X 1012
v = ( 6.67 X 1012)1/2  =  2.58 X 106 m/s