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Proof for projectile motion

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A body is thrown horizontally from a cliff with an initial horizontal speed. It hits the ground at a distance X from its initial position. It is said that the displacement = horizontal speed * time.

When the body falls, gravity acts upon it. So why don't we consider gravity ?

Any proof that the displacement would be the same as above if the same body would have moved with the same initial speed on a flat horizontal path for the same time period ?

Of course in the 2nd situation, it would continue to move infinitely.

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