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Gravitation

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Question : 20
Total: 25
Two stars each of one solar mass (=2×1030‌kg) are approaching each other for a head on collision. When they are at a distance of 109‌km, 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
Initial distance between two stars, r=109 km =1012 m
Initial potential energy of the system =−
GMM
r
Total K.E. of the stars =
1
2
Mv2+
1
2
Mv2=Mv2
wherev is the speed of stars with which they collide. When the stars are about to collide, the distance between their centres, r′=2R
∴ Final potential energy of two stars =−
GMM
2R
Since gain in K.E. is equal to loss in P.E.
∴Mv2=−
GMM
r
−(−
GMM
2R
)
=
−GMM
r
+
GMM
2R
⇒v2=
−GM
r
+
GM
2R
⇒v2=−6.67×10−11×(2×1030)(
1
1012
+
1
2×107
)
or v2=6.67×1012
⇒v=2.6×106ms−1
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