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 8.2 km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
Answer:
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 × 10-11 × 2 × 1030 [ (-1/1012 ) + (1 / 2 × 107) ]
~ 6.67 × 1012
v = ( 6.67 × 1012)1/2 = 2.58 × 106 m/s.
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