Energy question

Question

Determine W (or fuel energy) required to launch a satellite of mass m at rest from a launching pad placed at the surface earth, to a circular orbit where the radius of the orbit is r = 5 R where R is the radius of the earth. Ignore the effect of the Earth’s rotation. Take the mass of the Earth to be M. The fuel energy required is
Answer Choices:
1. 5GmM/ (4R)
2. 3GmM/(2R)
3. 7GmM/(6R)
4. 2GmM/R
5. 9GmM/(10R)
6. 5GmM/(6R)
7. 11GmM/(10R)
8. 3GmM/(4R)

Answer 1

r = 5R ⇒

 

v = √(GM/r) = √[GM/(5R)]

 

The total energy is

 

E = mv²/2 - GMm/r = GMm/(10R) - GMm/(5R) = -GMm/(10R)

 

On the surface of the earth at rest, the total initial energy is all gravitational potential energy.

 

E_i = -GMm/R

 

The work required to attain the orbit is 

 

W = E - E_i = -GMm/(10R) - (-GMm/R) = (9/10)GMm/R

 

It looks like answer (5) is correct.