If a bullet was fired perfectly straight up (assuming a windless environment) such that the peak of its trajectory was just shy of orbit, would it fall back to Earth in the same spot, or would Earth’s rotation cause it to land in a different place?
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What you describe is a thought experiment of course; it's not feasible to perform it in the real world. Still, we can consider it: the scenario does not include violation of any physics law.
For the thought experiment we remove any friction. So we have an Earth size planet, with Earth strength gravity, the same rotation as Earth, but no atmosphere.
First: in the case of a projectile that is fired there is no such thing as ending up in orbital motion. There are two possible outcomes: the projectile falls back to the planet, or the projectile escapes the planet's gravity. But no matter the speed or the angle of firing, ending up in orbital motion is not one of the possible outcomes.
The knife edge case is where the projectile has just enough velocity so that it does not fall back to the planet, but as it moves away from the planet its velocity (relative to the planet) drops off ever closer to zero.
Firing the projectile
For simplicity let's say the projectile is fired from the Equator, perfectly straight up. We can think of the initial velocity as the resultant velocity of two perpendicular contributions: a) the velocity of being fired straight up, b) the velocity of co-rotating with the Earth, along the Equator, prior to being fired.
First a low altitude case: the projectile climbs to a height that is considerably smaller than the planet's radius.
General statement:
Once the projectile is on its way its motion is the result of just a single force, the gravity of the planet. As we know, that gives a Kepler orbit, in this case a very elongated orbit. That is, the motion with respect to the Earth's center of gravity will be along an elongated ellipse.
The motion will be according to Kepler's law of areas: in equal amounts of time the orbit sweeps out equal areas. So the motion of the projectile will have the following property: the larger the distance to the center of attraction, the slower the angular velocity.
Therefore as the projectile climbs to its peak altitude the planet rotating underneath is at its fixed rotation rate, but during the flight the angular velocity of the projectile will at all times be slower than the planet's angular velocity. (And the angular velocity of the projectile will be at its slowest when it arrives at its peak altitude.)
So we see that the projectile will not land at the geographic spot that it was fired from. The projectile will impact the planet at a spot that is lagged behind.
The Earth turns from West to East: hence when the projectile impacts it will be at a spot westward from the point of firing. The longer the duration of the flight, the more westward.
In this thought experiment there is no limit to how much velocity can be imparted to the projectile. So you can give it so much velocity that it takes a full day to impact the planet again. Of course, by then it isn't practical anymore to refer to the impact spot as 'east' or 'west' of the starting location.
Concluding remarks:
Key to understanding what will happen is that after the projectile has been fired it is subject to only a single force: the planet's gravity. The subsequent motion is determined by two factors: the strength of the gravity, and the initial velocity (that includes the exact direction of that velocity.)
The rotation rate of the planet is a factor in the following sense: one contribution to the initial velocity is the velocity of co-rotating with the planet, prior to being fired. After being fired the subsequent motion of the projectile is independent of the rotation of the planet underneath it.
There is no need to pull in any other consideration, that would only introduce unnecessary complexity.
Cleonis
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Assuming the person who fired the gun was standing on the ground when the bullet was fired, and there are no outside forces acting on the bullet, no shooting into a bunch of birds, no wind AT ALL, etc. then it should land in roughly the same spot. (The person's aim also must be completely accurate)
The person standing on the ground(and thus, the gun/bullet) is already moving in the direction that the earth is rotating. The rotational energy of the bullet fired doesn't change just because upward momentum was applied. The bullet still travels in the direction of the earth's rotation since there are no other forces altering it's direction.
The bullet may gain and lose altitude, but the energy traveling in the direction of the earth's rotation is unaffected.
