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This question already has an answer here:

If moon travels with constant speed in one direction and earth gravitation causes constant acceleration in perpendicular direction why moon does not eventually fall into earth? I mean if gravitation causes moon to fall faster each second (10m/s2) shouldn't after time velocity toward earth be big enough to cause it to fall ?

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marked as duplicate by BMS, David Hammen, Floris, Ali, Kyle Oman Sep 10 '14 at 21:17

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    $\begingroup$ Certainly this is a duplicate of some other question! $\endgroup$ – David Hammen Sep 10 '14 at 19:45
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/9049/2451 and links therein. $\endgroup$ – Qmechanic Sep 10 '14 at 19:48
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    $\begingroup$ The value of g (10m/s2) doesn't apply to the moon, since that is Earth's surface gravity. It's about 1000 times smaller when you are as far away as the moon. $\endgroup$ – richardb Sep 10 '14 at 20:25
  • $\begingroup$ XKCD's related explanation: "To avoid falling...into the atmosphere, you have to go sideways really, really fast." $\endgroup$ – Justin Sep 10 '14 at 20:40
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It can be said that the tangential speed of the moon in its orbit is represented by a vector that is constant in magnitude, but not so his direction. This variation of the vector direction (always remains tangent to lunar orbit), is actually a change in velocity, and therefore acceleration.

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Why the moon does not fall on the ground? Simplifying to a circular orbit, the centripetal force acting on the moon's is the gravitational force, while by the movement itself, a centrifugal force that keeps the system in balance, is generated.

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    $\begingroup$ You could also explain that the gravitational force isn't so much pulling the moon to the earth as stopping the moon from flinging itself off into space. $\endgroup$ – PipperChip Sep 10 '14 at 19:20
  • $\begingroup$ @PipperChip Yes! It's a very good point of view! Tanks for your comment... ! $\endgroup$ – Martin Petrei Sep 10 '14 at 19:23
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Acceleration can change velocity in two ways - by changing its magnitude, and by changing its direction.

Essentially, Earth's gravity is constantly steering the moon around the Earth.

Your initial premise - "If moon travels with constant speed in one direction" - is incorrect. The moon's direction is constantly being changed by the gravitational acceleration.

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