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Gravity and Newton
The beautiful glowing Moon—our night-light with its specular brightness in various shades of gray—is held in orbit by the same forces that keep the planets and the comets in their orbits. It is the same force that enables the careful positioning of communications satellites encircling the Earth. That powerful force is gravity.
Newton’s Law of Gravitation states,
F = G (m1m2)/r2,
where F is the force, G is the gravitational constant (6.6730 x 10-11 m3/kg-s2), m1 and m2 are the masses of two interacting objects (in this instance, the Earth and Moon), and r is the distance between them.
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How Much Force?
The masses of the Earth and of the Moon are approximately 5.9736 x 1024 kg and 7.3477 x 1022 kg, respectively. The distance between Earth’s center and the Moon varies slightly, but is approximately 3.8440 x 108 m. This means the force due to gravity between the Earth and the Moon averages,
F = (6.6730 x 10-11)(5.9736 x 1024)(7.3477 x 1022) / (3.8440 x 108)2
where the units have been left off for the sake of space. The result is,
F = 1.9822 x 1020 kg-m/sec2 = 1.9822 x 1020 Newtons
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So What Keeps Them From Colliding?
When a space vehicle is launched with the goal of putting it in orbit about Earth, the vehicle reaches the proper distance from Earth by a trajectory that is carefully determined. Velocity is an important part of that. When the proper velocity and orbital path have been reached, and the craft is beyond Earth’s atmosphere, the vehicle orbits the Earth without falling into the Earth. This is because the vehicle falls around the Earth in a circular or more generally, an elliptical path. Enjoy the excellent article entitled “Basics of Orbital Mechanics and Interplanetary Trajectories" by Apollo scientist George Adcock for further details.
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What is the Moon's Velocity?
It is essentially the same situation that enables the Moon to remain in its orbit about the Earth. It has sufficient velocity to maintain its trajectory or “free fall" about the Earth. What is that velocity? Since the Moon revolves about the Earth in 27.322 days,
V = 2пr / t, or
V = 2 x 3.142 x (3.844 x 108 m) / (27.322 days x 24 hrs/day x 60 mins/hr x 60 secs/min)
V = 1.023 x 103 m/s = 2,288 mph
Much faster and it could escape Earth-orbit; much slower and it would crash into the Earth.
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Views of the Solar System - The Moon
NASA - Moon Fact Sheet