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 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? Since that is not far enough, you consult a friend who calculates (correctly) the minimum speed needed as vmin = 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! PLEASE SHOW ALL THE STEPS I NEED THAT TO UNDERSTAND THIS PROBLEM 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? Since that is not far enough, you consult a friend who calculates (correctly) the minimum speed needed as vmin = 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! PLEASE SHOW ALL THE STEPS I NEED THAT TO UNDERSTAND THIS PROBLEM 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? Since that is not far enough, you consult a friend who calculates (correctly) the minimum speed needed as vmin = 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! PLEASE SHOW ALL THE STEPS I NEED THAT TO UNDERSTAND THIS PROBLEM

Explanation / Answer

conservation of energy
Ei = Ef
KE + PEearth + PEmoon = PEearth + PEmoon

1/2 mv^2 - G m Mearth/(r) - G m Mmoon/(d-r) = -G m Mearth/x -G m Moon/(d-x)

0.5*5534^2 - 6.67E-11*5.9742E24/6.3781E6 - 6.67E-11*7.36E22/(3.844E8-6.3781E6)=- 6.67E-11*5.9742E24/x - 6.67E-11*7.36E22/(3.844E8-x)

x=8.45E6 m

2)

now using your friends velociy

1/2 mv^2 - G m Mearth/(r) - G m Mmoon/(d-r) = 1/2 mv^2 -G m Mearth/(d-rmoon) -G m Moon/(rmoon)

0.5*11068^2 - 6.67E-11*5.9742E24/6.3781E6 - 6.67E-11*7.36E22/(3.844E8-6.3781E6)=0.5*v^2-6.67E-11*5.9742E24/(3.844E8-1.7374E6) - 6.67E-11*7.36E22/(1.7374E6)

v=2293 m/s

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