Question

Two stars each of one solar mass (= 2 ×10³⁰ kg) are approaching each other for a head on collision. When they are at 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. (Used the known on value of G).

Answer 1

given, Mass of each star, m = 2 × 10^30 kg
Radius of each star, r = 10⁴ km = 10^7 m
Distance between the stars, R = 10^9 km = 10¹² m

For negligible speeds, v = 0
total energy of two stars separated at distance r = potential energy + kinetic energy
= -Gmm /R  + 1/2mv2
=  -Gm² / 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 + 1/2mv²  = mv²

Total potential energy of both stars = -Gmm / (r + r) = -Gm²/2r

Total energy of the two stars = mv² – Gm² / 2r   ...….(ii)

From law of conservation of energy,
Total energy initially = total energy finally.
mv² – Gm²/2r =  -Gm²/R

v² = -Gm/r + Gm/2r

= GM { (-1/R) + (1/2r) }

= 6.67 × 10^-11 × 2 × 10^30 [ -1/10¹² + 1 / 2 × 10^7 ]

V² ≈ 6.67 × 10¹²
v = ( 6.67 × 10¹²)½  =  2.58 × 10^6  m/s.

Answer 2

Answer:

2.58×10⁶m\s

this is the answer