Question

You plan to take a trip to the moon. Since you do not have a traditional spaceship with rockets, you will need to leave the earth with enough speed to make it to the moon. Some information that will help during this problem:

m_earth = 5.9742 x 10²⁴ kg
r_earth = 6.3781 x 10⁶ m
m_moon = 7.36 x 10²² kg
r_moon = 1.7374 x 10⁶ m
d_{earth to moon} = 3.844 x 10⁸ m (center to center)
G = 6.67428 x 10^-11 N-m²/kg²

1)

On your first attempt you leave the surface of the earth at v = 5534 m/s. How far from the center of the earth will you get?

2)

Since that is not far enough, you consult a friend who calculates (correctly) the minimum speed needed as v_min = 11068 m/s. If you leave the surface of the earth at this speed, how fast will you be moving at the surface of the moon? Hint carefully write out an expression for the potential and kinetic energy of the ship on the surface of earth, and on the surface of moon. Be sure to include the gravitational potential energy of the earth even when the ship is at the surface of the moon!

Answer 1

Both of these parts can be solved by carefully applying energetics. In the first case, you move with some initial kinetic energy based on the speed given. You start off at the surface of the Earth, which is a distance r (earth) away from its center. In this case, I have chosen to call the center of the Earth the origin since we want to know how far away the shuttle is from there. Now, for the final state, since the shuttle doesn't make it far enough, it must have stopped at some distance, right? So, the final kinetic energy is zero, and you are some distance away from the Earth. All you need to do is solve for that distance using conservation of energy.

The second problem involves the same equations with a new initial speed. You still leave from the surface of the Earth, but manage to JUST make it to the surface of the moon. If you barely make it, that means you barely land with no speed (zero kinetic energy). This time though, the distance is given by the center to center distance betwenn the Earth and moon, minus the radius of the moon. (Again, this distance is measured from the center of the Earth, although you could choose any reference point).

Hope this helps!