Answer: 1010.92 m/s
Explanation:
According to Newton's law of universal gravitation:
[tex]F=G\frac{Mm}{r^{2}}[/tex] (1)
Where:
[tex]F[/tex] is the gravitational force between Earth and Moon
[tex]G=6.674(10)^{-11}\frac{m^{3}}{kgs^{2}}[/tex] is the Gravitational Constant
[tex]M=5.972(10)^{24} kg[/tex] is the mass of the Earth
[tex]m=7.349(10)^{22} kg[/tex] is the mass of the Moon
[tex]r=3.9(10)^{8} m[/tex] is the distance between the Earth and Moon
Asuming the orbit of the Moon around the Earth a circular orbit, the Earth exercts a centripetal force on the moon, which is equal to [tex]F[/tex]:
[tex]F=m.a_{C}[/tex] (2)
Where [tex]a_{C}[/tex] is the centripetal acceleration given by:
[tex]a_{C}=\frac{V^{2}}{r}[/tex] (3)
Being [tex]V[/tex] the orbital velocity of the moon
Making (1)=(2):
[tex]m.a_{C}=G\frac{Mm}{r^{2}}[/tex] (4)
Simplifying:
[tex]a_{C}=G\frac{M}{r^{2}}[/tex] (5)
Making (5)=(3):
[tex]\frac{V^{2}}{r}=G\frac{M}{r^{2}}[/tex] (6)
Finding [tex]V[/tex]:
[tex]V=\sqrt{\frac{GM}{r}}[/tex] (7)
[tex]V=\sqrt{\frac{(6.674(10)^{-11}\frac{m^{3}}{kgs^{2}})(5.972(10)^{24} kg)}{3.9(10)^{8} m}}[/tex] (8)
Finally:
[tex]V=1010.92 m/s[/tex]
