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A bullet is fired horizontally from a gun at exactly the same time that a similar bullet is allowed to fall to the ground from the same height....

This is the sort of question that is better answered through experiment than theory, since it's fairly simple to test and would provide an actual answer instead of a hypothesis. I've linked a clip from a television program in which this experiment was conducted, and confirmed the prediction that we would derive from math: both bullets hit the ground at the same time.


This probably seems counter-intuitive because we think that, by traveling so much faster, the fired bullet must have some kind of different property relating to its vertical speed. In fact, the horizontal speed of the fired bullet is a completely different property, unrelated to gravity, that only determines how far the bullet is able to travel in a given amount of time. This is why muzzle velocity and elevation are so important in determining the strategic uses of various firearms.


Mathematically, we could use the equation X = Vi - .5at^2 to solve this;


The initial speed of both bullets is 0 (they are at rest relative to the ground), so the equation simplifies to X = -.5at^2


a is 9.8m/s on average, so half of that becomes 4.9


We could assign any value to X, so in this example I'll say it's 1.5 meters, the height of a reasonably sized person.


1.5 = 4.9t^2


therefore, t = 0.55 seconds


This seems like a pretty short timespan, but considering that a modern gun can fire over 1000m/s, this means the fired bullet will be over 500 meters, or 1500 feet, away from you by the time it hits the ground.

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