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

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 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!

3)

Which of the following would change the minimum velocity needed to make it to the moon?

the mass of the earth

the radius of the earth

the mass of the spaceship

Explanation / Answer

(1) According to the law of conservation of energy
initilal energy = final energy
Initial energy at the earth surface = potential energy + kinetic energy
= -GMem/re + (1/2)mv2
Final energy = potential energy at the height h above the earth surface
Final energy = -GMem/re+h
Now,
-GMem/re + (1/2)mv2  = -GMem/re+h
where G is the gravitational constant , Me is the mass of earth , m is the mass of the spaceship and re is the radius of earth
(6.67*10-11*5.9742*1024)/ 6.3781*106   + (1/2)(5534)2 = (6.67*10-11*5.9742*1024)/ (6.3781*106 + h)
on solving for h
h = 2.07*106 m
(2) Let us consider that the velocity at the moon is u
now applying same energy conservation
  -GMem/re + (1/2)mv2 =  -GMmm/rm +  (-GMem/d) + (1/2)mu2
where v = 11068 m/s , Mm is the mass of the moon , d is the distance between the earth and the moon.
on solving we get
u = 2296 m/s
(3) the minimum velocity needed to make it to the moon or escape the earth is given by
Vm = (2GMe /re)1/2
hence the minimum velocity depends on the value of the mass and radius of the earth.

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