Object in elliptical orbit around another object

[mybrohshi5]

Homework Statement

In a science fiction story, humans have built a large spherical spaceship for an interstellar journey of many generations; they have christened their ship Big Bird.

The mass of Big Bird is 7.496×1011 kg (the mass of an aircraft carrier is about 10^8 kg). Most of Big Bird's journey takes place at constant speed through the vacuum of interstellar space.

Big Bird is accompanied on its journey by a small satellite which is held in orbit about Big Bird solely by Big Bird's gravitational attraction. The satellite is called Little Bird and has a mass of 1000 kg.

Little Bird is in an elliptical orbit about Big Bird; the distance of closest approach is 5 km and the greatest separation distance is 40 km.

Find the difference in total mechanical energy for the system; i.e. mechanical energy at maximum separation minus mechanical energy at closest approach.

Homework Equations

E = K + U

K = 1/2mv²

U = -Gmm/r

The Attempt at a Solution

I am a little confused on this problem. i only seem to know the equation for V for a circular orbit so i am not sure how to go about this problem.

would i maybe do this m_L = mass of little bird and m_B = mass of big bird

1/2m_Lv² + -Gm_Bm_L / r₁ - [1/2mv² + -Gm_Bm_L / r₂ ]

r₁ = 40,000 m
r₂ = 5,000 m

Thanks for any suggestions on what to do cause i am completely lost on how to go about this problem.

 

[ideasrule]

Mechanical energy should be conserved because no forces are acting on the system, shouldn't it?

 

[mybrohshi5]

Yes it should be conserved.

Am i even on the right track to find the difference in total mechanical energy for the system?

I am a little confused on what equations to use i guess.

I thought i should use:

K1 + U1 - (K + U)

Where K1 and U1 are:

1/2m_Lv² and Gm_Bm_L / r

r = 40,000 m

K and U are:

1/2m_Lv² and Gm_Bm_L / r

r = 5000 mand v can be found using Gm_B / r

r = 40,000 for the K1 and r = 5000 for K

Am i on the right track with these equations and using the right "r" for each equation or am i not even close?

Thank you