Two satellites, of masses m and 3m, respectively, are in the same circular orbit

Question

Two satellites, of masses m and 3m, respectively, are in the same circular orbit about the Earth's center, as shown in the diagram above. The Earth has mass Me and radius Re. In this orbit, which has a radius of 2Re, the satellites initially move with the same orbital speed vo but in opposite directions.

photo of scenario: http://tinypic.com/view.php?pic=xkrx9u&s=9#.VxYdLMfKe8U

a. Calculate the orbital speed v0 of the satellites in terms of G, Me, and Re.

b. Assume that the satellites collide head-on and stick together. In terms of v0 find the speed v of the combination immediately after the collision.

c. Calculate the total mechanical energy of the system immediately after the collision in terms of G, m, Me, and Re. Assume that the gravitational potential energy of an object is defined to be zero at an infinite distance from the Earth.

Explanation / Answer

a)

we know that GMe m/4Re^2= mv^2/2Re

V0= sqrt (GMe/2Re) for both satelites

b)

now conserving momentum at the time of collision

mV0 + 3m V0 = 4m ( V)

V = V0

c)

E = -GMe(4m)/2Re + GMe 4m / 4Re

E =- GMe m / Re

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