Air density drops with increasing altitude. The object dropped from a higher altitude reaches a higher speed before reaching the denser air where object B is dropped. I’m not sure if a realistic density profile will allow object A to arrive first, but it is easy to show that there is some air density profile which will cause this to happen. I suspect that a necessary condition is that object A is already above the terminal velocity at object B’s initial height when it reaches that height.
Or, if you interpret “free fall without any initial relative velocity against the planet” to say that it is stationary with respect to both the Earth’s center of mass and the Earth’s surface, then drop B from a geostationary orbit, and A from a higher position, where it will have insufficient angular velocity to be in orbit. It will fall to Earth, while B’s orbit will decay.
Edit: It is permitted to assume that they are dropped over the equator, since the problem says “Central Atlantic”.
Edit 2: Wait, I did this wrong. If object A has a rotational velocity of 1/day, and it is at an altitude higher than a geostationary orbit, it will be in some larger more eccentric orbit, so it won’t fall to Earth any sooner than object B.
It’s not an escape orbit, it’s just a more eccentric orbit (unless it is much higher). Still, you are correct that my second solution will not work (see my second edit).
I started solving the trajectory for an exponentially decaying air density and a drag force that scales linearly with density and quadratically with velocity, but I did not immediately see the solution to the resulting differential equation, nor did I see a clever trick for avoiding the calculation. I’ll look at it again later.
Air density drops with increasing altitude. The object dropped from a higher altitude reaches a higher speed before reaching the denser air where object B is dropped. I’m not sure if a realistic density profile will allow object A to arrive first, but it is easy to show that there is some air density profile which will cause this to happen. I suspect that a necessary condition is that object A is already above the terminal velocity at object B’s initial height when it reaches that height.
Or, if you interpret “free fall without any initial relative velocity against the planet” to say that it is stationary with respect to both the Earth’s center of mass and the Earth’s surface, then drop B from a geostationary orbit, and A from a higher position, where it will have insufficient angular velocity to be in orbit. It will fall to Earth, while B’s orbit will decay.
Edit: It is permitted to assume that they are dropped over the equator, since the problem says “Central Atlantic”.
Edit 2: Wait, I did this wrong. If object A has a rotational velocity of 1/day, and it is at an altitude higher than a geostationary orbit, it will be in some larger more eccentric orbit, so it won’t fall to Earth any sooner than object B.
Your first idea should be elaborated. But it is quite sound.
Your second idea is wrong. Above geostationary orbit and not moving relative to the Earth surface, means an escape orbit.
Still, I would prefer a non—atmosphere solution. But yes, your first idea is also good albeit a little undeveloped.
It’s not an escape orbit, it’s just a more eccentric orbit (unless it is much higher). Still, you are correct that my second solution will not work (see my second edit).
I started solving the trajectory for an exponentially decaying air density and a drag force that scales linearly with density and quadratically with velocity, but I did not immediately see the solution to the resulting differential equation, nor did I see a clever trick for avoiding the calculation. I’ll look at it again later.
You are absolutely right. It CAN be an escape orbit, if it is high enough. But it may also not be an escape orbit.
You are right. Still, not a good solution.