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Projectile Launched around the Earth

t
y
0
v
x
0
Suppose that a projectile is launched at a height
y
0
from the surface of the Earth (idealized here as a perfect sphere). If the initial velocity (
v
x
0
) or the height is not large enough, the projectile can not travel very far during the time
t
, subject to its gravitational attraction. However, with a larger velocity, the projectile can travel a considerable distance around the Earth, and even remain in elliptical orbit if the velocity is large enough.
Pictures illustrating these facts were published in Newton's Principia.
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