Is it true that if a bullet is dropped from the same height and at the same time a bullet leaves a rifle that they will both hit the ground at the same time?
What are the formulas for proving this?
Yes.
You don’t really need a formula to prove this, it is just obvious from the fact that the horizontal component of the bullet’s velocity can be ignored, since gravity only acts in the vertical direction.
The bullet will accelerate downwards under gravity at about 9.8 m/s[sup]2[/sup], regardless of how fast it is travelling forwards.
Is there a minimum height that they must begin at in order for this to remain true?
Well, the caveat would be that the rifle is fired horizontally, and that you neglect air resistance.
Then, you break down the velocity and acceleration factors into horizontal and vertical components.
With regards to the rifle: In the horizontal direction, you have a certain velocity v[sub]h[/sub], but no acceleration a[sub]h[/sub] because there’s no force acting on the bullet. So the bullet’s horizontal velocity never increases or decreases.
Meanwhile, in the vertical direction, you initially have zero velocity v[sub]v[/sub], but you have acceleration a[sub]v[/sub] that is 9.8 meters per second squared (what is that, 32 feet per second squared?) downward. Whether you consider up or down to be in the ‘negative acceleration’ direction is just by convention.
Because of this acceleration, it takes a certain amount of time to hit the ground based on the height it is originally fired at. It has to cover the distance d, which, given that the initial velocity in that direction is zero, is t = sqrt(2d/a[sub]v[/sub]).
This time top drop is independant of the fact that it happens to also be going horizontally at some rate.
With regards to the dropped bullet: The exact same arguments can be made about the time to drop from the same height. Although there is no horizontal velocity, we’ve already covered the fact that the horizontal velocity is irrelevant. The time to drop is exactly the same formula based on the distance d to the ground, and the acceleration a[sub]v[/sub] because the acceleration due to gravity is the same.
Because we’ve not put any caveats on the distance above ground, the height is irrelevant, until you start having to take into account curvature of the Earth for the horizontal travelling bullet… but in a localized ‘flat Earth’ problem, height doesn’t matter. You’re assuming flat and level ground anyway.
Thank you for taking the time to help me understand this. I will still have to go over it a couple of times to let it all sink in, but you have very helpful.