Getting to the moon is tricky business. Buzz, Mike and Neil had to go fast enoug

Question

Getting to the moon is tricky business. Buzz, Mike and Neil had to go fast enough to get away from Earth’s gravity, but slow enough to not overshoot the moon (there are no brakes on spacecraft!) Assuming no thrusters were used to slow the Command Module on its way to the moon, find the slowest possible speed it could have been moving as it reached a point 500 km from the moon’s surface. The distance from Earth to moon (center to center) is 384,000 km. (Hint: see Webassign Problem 2. And draw a good picture.)

Note: the answer to Problem 4 is greater than the speed you found in Problem 2, which tells us that the astronauts must have used thrusters to slow the command module as it approached the moon.

mass of Earth=5.98x1024 kg radius of Earth=6.37x106 m mass of sun=1.99x1030 kg

Can you please give a detailed explanation of how you got the answer including any free body diagrams and formulas you used. The answer is 1980 m/s.

Explanation / Answer

Mass of the moon,

m=7.34*10^22 kg

Escape velocity of the moon, at r=500+radius of the moon = 500+1737 km =2237*10^3 m

vescape = root [2*G*m/r]

=root[2*6.67*10^-11*7.34*10^22/2237*10^3]

=2092 m/s

vescape_earth = root [2*G*M/R] R =384,000-2237 381763 (min)

=root[2*6.67*10^-11*5.98x10^24/381763*10^3]

=2092 m/s

Veclocity has to be less than escape velcoity of the moon, so maximum veclocity the speacecraft can afford, is 2092 m/s.

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