Satellite Collision: Find Max Mass mj for Orbit Without Hitting Earth

[jhoge]

Homework Statement

A satellite of mass m is in a circular orbit of radius 2RE around the Earth. The satellite hits a piece of space junk of mass mj. The space junk, which was initially at rest, sticks to the satellite. The satellite and junk then fall as one in a new elliptic orbit around the Earth. What is the maximum value of the junk mass mj such the satellite of m is to remain in orbit without hitting the Earth? Write your expression in terms of m and no other variables or constants.

Neglect the Earth's atmosphere. Explicitly indicate multiplication using the * symbol

Homework Equations

The equations that are relevant to this problem are the conservation of angular momentum and the conservation of energy equations.

In this case the conservation of angular momentum for the satellite before the collision is :
m*Vo*2R. (Vo is the initial orbital velocity of the satellite. Considering the centripetal
force acting on the satellite we can determine its orbital velocity to be √(G*M/R) At aphelion, the angular momentum of the combined mass of satellite and space junk is:
(m+mj)Va*2R (I am considering the collision to occur at aphelion as this is the farthest point
the satellite can reach from Earth. The satellite slows upon collision, resulting in the elliptical orbit.

At perihelion, the angular momentum is:
(m+mj)Vp*R

The Attempt at a Solution

My approach involves using the conservation of energy and momentum equations mentioned above. In short, I used to conservation of angular momentum to say that

(1) m*Vo*2R = (m+mj)Va*2R &
(2) m*Vo*2R = (m+mj)Vp*R

Using the conservation of energy equation (energy is only conserved after the collision since it is inelastic):

(3) (1/2)(m+mj)(Vp)^2 - GM(m+mj)/R = (1/2)(m+mj)(Va)^2 - GM(m+mj)/2R

I get : (3) Vp^2 - Va^2 = GM/R

Solving for Vp using equation (1) i get Vp = 2*m*√(G*M/R)/(m+mj)
Solving for Va using equation (2) i get Va = m*√(G*M/R)/(m+mj)

Plugging these values into (3) i get

(4) 4m^2Vo^2/(m+mj)^2 - m^2Vo^2/(m+mj)^2 = G*M/R.

Given that Vo = √(G*M/R), equation (4) simplifies to 3m^2 = (m + mj)^2, which leaves me with a value of

mj= (√(3)-1)*m.

I would like to know if this approach is the correct one, because I have one submission attempt left. Thanks in advance.

 

[jhoge]

forgot to include photo

here is a photo