Physics (orbits and energy)?

Question

Any help on this physics homework problem is appreciated. I've tried it and have no idea how to properly relate the energy equations to this scenario.

An engine-less space capsule is launched from the surface of the Earth and travels into space until it experiences essentially zero gravitational force from the Earth. The initial speed of the capsule is 18,500 m/s. What is its final speed? Assume no significant gravitational forces from other solar system bodies. Mass of Earth = 5.97e24 kg. Radius of Earth = 6.38e6 m.

Answer 1

If teh capsule experiences no gravitational force, then it is so far away that you can consider it at infinity. The work needed to overcome the gravitational potential energy of the earth and move an object to infinity is:

W = GMm/Re where Re = radius of earth. If you have this much energy you arrive at infinity with no kinetic energy at all - hence you don't move. The equivalent kinetic energy is:

K = 1/2mv^2 = GMm/Re ---> vesc = sqrt(2*GM/Re) = 11.1 x 10^3 m/s = escape speed

Now the capsule has vc = 18.5x10^3 m/s initial speed. It has an initial KE of 1/2 m vc^2. So the speed at infinity is found using conservation of energy:

1/2 m vc^2 = 1/2mv^2 +1/2 m vesc^2 this simplifies

vc^2 = v^2 + vesc^2 ---> v = sqrt(vc^2 - vwsc^2) = 14.8x10^3 m/s