You plan to take a trip to the moon. Since you do not have a traditional spacesh

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:

mearth = 5.9742 x 1024 kg
rearth = 6.3781 x 106 m
mmoon = 7.36 x 1022 kg
rmoon = 1.7374 x 106 m
dearth to moon = 3.844 x 108 m (center to center)
G = 6.67428 x 10-11 N-m2/kg2
v = 5534m/s

Since that is not far enough, you consult a friend who calculates (correctly) the minimum speed needed as vmin = 11068.0 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!

Explanation / Answer

Gravitational potential energy is U U = -GMn/r where G is the universal gravitational constant, M is the mass of Earth, m is the mass of the spaceship and r is distance form the center of the Earth. Kinetic energy is E E= 1/2mv^2 where m is the pass of the space ship and v is the velocity. Since U is negative and gets smaller as you go higher, you can just calculate the E at launch, calculate U at the surface of Earth, then add them together. The new U will still be negative but a smaller absolute value (unless you launch at escape velocity or faster; it will be 0 with launch at escape velocity and positive if launched faster). Then you use the new U, combined with the known G, M, and m to calculate the new r. To find the speed at a certain distance, calculate U at that distance and add the launch E. Then set the resulting new U equal to E and calculate v. The mass of the space ship makes no difference in the minimum velocity needed to reach the Moon, but mass and radius of Earth do.

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