Given:
- Two uniform spherical stars made of the same material
- Radii of the stars: R and 2R
- Mass of the smaller star: m
- They start moving from rest towards each other from a large distance under mutual force of gravity
- The collision between the stars is inelastic with a coefficient of restitution 1/2

To find:
The maximum separation between their centers after their first collision

Explanation:
When the stars start moving towards each other, they experience a mutual gravitational force that accelerates them towards each other. Let's analyze the motion of the stars before and after the collision.

Before the collision:
- The stars are at a large distance from each other and are moving towards each other due to the gravitational force between them.
- Since the stars are initially at rest, their initial velocities are 0.

First collision:
- The stars collide inelastically, meaning they stick together after the collision.
- In an inelastic collision, the kinetic energy is not conserved, but the momentum is conserved.

After the collision:
- The stars stick together and move as a single object.
- The final velocity of the combined object can be found using the principle of conservation of momentum.

Conservation of momentum:
- The total momentum before the collision is zero since both stars are initially at rest.
- The total momentum after the collision is the sum of the individual momenta of the stars.
- Let the final velocity of the combined object be V.
- Using the conservation of momentum, we can write:
0 = mv1 + 2m*v2 (1)
where v1 and v2 are the initial velocities of the smaller and larger star, respectively.

Coefficient of restitution:
- The coefficient of restitution (e) is defined as the ratio of relative velocity after collision to the relative velocity before collision.
- In this case, the relative velocity before collision is v2 - v1.
- The relative velocity after collision is (V - v1) - (V - v2) = v2 - v1.
- Using the given coefficient of restitution (e = 1/2), we can write:
(v2 - v1) / (v2 - v1) = 1/2
2(v2 - v1) = v2 - v1
v2 = 3v1 (2)

Maximum separation between their centers:
- After the collision, the stars move as a single object with velocity V.
- The maximum separation between their centers occurs when the stars come to rest.
- At rest, the velocity of the combined object is 0.
- Using equation (2), we can write:
V = 3v1
0 = V - v1
v1 = V

Maximum separation:
- The maximum separation between their centers can be found by calculating the distance traveled by the smaller star.
- The distance traveled by the smaller star is given by:
d = v1 * t
where t is the time taken for the stars to come to rest.
- The time taken for the stars to come to rest can be found by analyzing the motion of the combined object using the principle of conservation of mechanical

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