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Shooting and dropping a bullet
If you would shoot a bullet from a gun exactly parallel to the Earth's surface, the motion of the bullet would have no effect on how gravity acts on the bullet. In other words, the bullet would drop at the same rate as a stationary object.
Dropped bullet and shot bullet hit the ground at the same time
Many people don't believe that if you held a rifle or handgun parallel to the ground and at the same time you shot the bullet, you dropped another bullet from the same height, both bullets would both hit the ground at the same time. However, it is a fact.
Exception
An exception to this phenomenon would be if the bullet or object was able to travel so many miles or kilometers that the curvature of the Earth came into play. In such a situation, the bullet would take slightly longer to hit the ground, because the displacement to the ground was greater due to the Earth's curvature.
Example
If you shot a bullet at 900 m/s from a rifle that was 1.5 m above the ground, how far would the bullet fly until it hit the ground? Discount air resistance and assume the rifle is parallel to the ground.
Solution
x = vs√(2y/g)
x = (900 m/s)√[2*(1.5 m)/(9.8 m/s2)]
x = (900 m/s)√(0.306 s2)
x = (900 m/s)(0.553 s)
x = 498 m or 1634 ft
Summary
An object moving sideways or parallel to the Earth's surface will fall at the same rate as one that is simply dropped. The equation for the displacement of the object before it hits the ground can be derived from the gravity equations for falling objects. An exception is if the object moves so fast or far that the curvature of the Earth comes into play during its fall to the ground.