I need help with a physics problem please

Question

Calculate the minimum amount of work needed to move a 400 kg rocket from Earth's surface to the International Space Station in orbit 300,000 m above Earths surface.

1. Solve for the minimum amount of work needed to move the payload to the ISS. Start with general equations and explain what you are doing and why you did what you did.
2. Evaluate your solution. Describe how you evaluated your answer and whether or not you think your answer is reasonable.

Answer 1

E₁ is the mechanical energy at the earth surface

E₂ is the mechanical energy at the space station

W is the work done on the rocket

G is the gravitational constant= 6.6743x10^-11 N.m²/kg²

R_e is the earth radius= 6378000 m

m is the rocket's mass= 400kg

Me is the mass of the earth=5.972x10²⁴kg

Due to conservation of Mechanical Energy E₁+W=E₂ , K₁+U₁+W=K₂+U₂

At the earth's surface, before the work is applied to the rocket, it is at rest, thus K₁=0

U₁= -GmM_e/R_e

At the space station the balance of force on the rocket says that the centripetal force should be equal to gravitational force. Thus:

mv₂²/(R_e+h)= GmM_e/(R_e+h)²

Thus: mv₂²= GmM_e/(R_e+h) and K₂=1/2mv₂²= GmM_e/2(R_e+h)

Thus K₂=GmM_e/2(R_e+h)

U₂= -GmM_e/(R_e+h)

From the conservation of Mechanical Energy U₁+W=K₂+U₂

Thus: W=GmM_e/R_e - GmM_e/2(R_e+h)

W = 13,060,378,071 J =1.3x10¹⁰ J