Actually, I think I’ve come up with a more elegant idea than altitude-triggered airbraking. There is no requirement that when released they go down into the gravity well :-D
Are the two objects interchangeable—that is, can we swap A and B and get the same result?
A simple answer to your question is that atmosphere is not homogeneous. Dropping one object, say, over a hurricane downdraft and another over an updraft would result in different velocities relative to ground and so the one dropped from a higher altitude can reach the surface earlier.
So it seems we can swap both the objects and the locations?
In your setup, if we take object A and lower it to the altitude of B, then take B and lift it up to the (former) altitude of A, will the object B reach ground first?
So it seems we can swap both the objects and the locations?
Yes.
In your setup, if we take object A and lower it to the altitude of B, then take B and lift it up to the (former) altitude of A, will the object B reach ground first?
This seems like it could have many possible answers. The most obvious is that A has a different shape or weight to B, and so is less affected by air resistance. Reading the problem literally it could just also be that A was dropped substantially before B.
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.
Given only that both objects will splash into the Central Atlantic, there is wiggle room in choosing latitudes and longitudes such that lateral distance between the objects is vastly larger than the difference in altitude. Suppose the lower-altitude object is placed safely inside the eastern boundary of this region and the higher-altitude object is placed inside the western boundary, such that each will splash into the Central Atlantic according to the rules. The objects can be several 1000km apart but only 1cm different in altitude.
Now we just need a natural force to slow the descent of the lower-altitude object more than it slows the higher-altitude object. Let’s use the gravity of the Moon. Drop the objects when the moon passes directly above the lower-altitude object. The Moon exerts a greater upward pull on the lower-altitude object, slowing its descent slightly more than it slows the descent of the higher-altitude object.
You don’t seem to prefer aerodynamic solutions but they are abundant so here’s another one.
Orientation is unspecified so we can drop one object right-side-up and drop the other upside-down. A shape can be designed such that both orientations are stable (neither will switch orientations during free-fall) and such that one orientation has a different ballistic coefficient, creating a different terminal velocity. This could be accomplished with fan blades or pitot tubes of different sizes or protruding at different angles, converting different amounts of kinetic energy to heat depending on the direction of airflow.
The thing that brings my attention is the phrase “without any initial relative velocity against the planet”. They might not move relative to the planet, but Earth is not an inertial reference frame and is rotating, so bodies in different latitudes would have different speeds in relation to an inertial referential.
My first thought was that B reaches terminal velocity and that’s it, but the object A is dropped from substantially higher altitude, picks up speed much higher than the terminal velocity and the atmosphere won’t slow it down enough,
I do not feel like oing the math, but there is a simpler solution:
jnvg sbe n jnir naq gvzr gur qebc va fhpu n jnl gung bowrpg N, orvat uvture ol yrff guna gur jnir urvtug, uvgf gur perfg bs gur jnir.
No, they both rotate or neither one rotates. They are equal, so the same rotation must be assumed. Their initial places are different and any difference only comes from that fact.
If both rotate (I assume the same angular velocity) , what can be said about the direction of their axes of rotation?
Another idea: Gurl ner zntargvp, pybfr rabhtu gung gurl nggenpg gurzfryirf, fgvpx gbtrgure naq gur ubevmbagny nflzrgel pnhfrf gur bowrpgf gb ghzoyr naq gur uvture bar pbzrf qbja svefg.
Two giant golden balls, dropped somewhere bellow the geosynchronous orbit might do the trick of a little orbiting around each other and then splashing into the ocean, one after another.
That might cause some damage, but the Earth would survive as a planet. Rather costly and not environment friendly solution.
Or two ordinary small balls, one dropped from just above the geosynchronous orbit, the second one from far above the orbit. While the first one slowly drifts away to the space, the second shoots away, makes a complete (retrograde) orbit around Sun and splashes into the Atlantic while the first ball is still drifting...
This is true, but those Moon or Sun solutions aren’t my favorite. Moon, Sun, Jupiter and so on are external agents I’ve forgotten to explicitly forbid. Next time, I’ll be even more careful when posting a problem. :-)
Airplanes dropped in different orientations, or in a way that’s sensitive to initial conditions and leads to B gliding while A stalls. B is dropped from right above a passing eagle and gets carried off to Mordor. They’re “dropped” far out past geostationary orbit so that “stationary relative to the planet” in fact means that they’re flung off into space, and only reach Earth by getting slingshotted around other planets. Both are dropped over Brazil with notes to please throw them into the Atlantic, and A is dropped so that its coriolis motion as it falls will push it to a more visible area. They’re microscopic black holes dropped from the other side of the earth.
Since you clearly have something in mind, once you reveal it, are we going to go “Oh, yeah, that’s much more sensible than the gliders that are dropped near the boundary of glide vs. stall air pressure,” or are we going to go “well, that’s arbitrary.”
Rigid hot-air-balloon shapes that start out at 1500 Celsius and fall to earth once they are no longer keeping the air under them hot. Seed crystals in a hailstorm. Any object that falls faster if broken and will break if dropped from the higher altitude. Solid-state electrostatic thrusters pointed downward, that arc and fail if the pressure is too high. Spinning propeller craft thrusting downward that undergo a laminar to turbulent transition if the pressure is too high.
The problem specified “freefall”, so thrusters are out. But I agree that it’s underspecified—there are way too many things which fit so we are reduced to guessing what Thomas had in mind.
The initial conditions did not forbid dropping object A into a downdraft and object B into an updraft :-)
But here is another attempt: objects A and B open parachutes 3 seconds after passing altitude X. Object B starts at altitude X, accelerates from zero for three seconds, and then radically slows down. Object A starts higher, so when it passes altitude X it is already going fast and so in three seconds is capable of passing B which is already braked by a parachute.
If this scenario is actually possible then it seems like the reason should be that the thing dropped from higher up gets to accelerate faster at first due to the thinner atmosphere. However, I’ve tried quite a lot of toy examples with different dependencies of air resistance on height and on speed, and none of them showed the trailing object ever quite catching up; unless I’ve screwed up or there’s some other thing going on (relating perhaps to the earth’s magnetic field or solar wind or something, but these seem like desperate longshots), either this doesn’t actually happen or it happens only when the circumstances are just right.
Another problem to solve:
https://protokol2020.wordpress.com/2017/02/05/physics-problem/
Actually, I think I’ve come up with a more elegant idea than altitude-triggered airbraking. There is no requirement that when released they go down into the gravity well :-D
Your two objects are leaky zeppelins.
Helium Zeppelins aren’t exactly rigid bodies. But vacuum Zeppelins are. Those could be arranged to do the job.
What does “equal” mean?
The same shape, mass and of the same material.
Are the two objects interchangeable—that is, can we swap A and B and get the same result?
A simple answer to your question is that atmosphere is not homogeneous. Dropping one object, say, over a hurricane downdraft and another over an updraft would result in different velocities relative to ground and so the one dropped from a higher altitude can reach the surface earlier.
Yes. We can swap A and B and get the same result.
No, we drop them both over the Central Atlantic and then both will have about the same weather.
It’s the same, unexceptional weather for both.
So it seems we can swap both the objects and the locations?
In your setup, if we take object A and lower it to the altitude of B, then take B and lift it up to the (former) altitude of A, will the object B reach ground first?
Yes.
Yes, the object B will reach ground first.
Can the objects do anything during their descent, such as opening a parachute at a predetermined altitude? changing their aerodynamics?
Also, can they include sensors and electronics?
This seems like it could have many possible answers. The most obvious is that A has a different shape or weight to B, and so is less affected by air resistance. Reading the problem literally it could just also be that A was dropped substantially before B.
No, they are equal.
No. At the same time.
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.
Given only that both objects will splash into the Central Atlantic, there is wiggle room in choosing latitudes and longitudes such that lateral distance between the objects is vastly larger than the difference in altitude. Suppose the lower-altitude object is placed safely inside the eastern boundary of this region and the higher-altitude object is placed inside the western boundary, such that each will splash into the Central Atlantic according to the rules. The objects can be several 1000km apart but only 1cm different in altitude.
Now we just need a natural force to slow the descent of the lower-altitude object more than it slows the higher-altitude object. Let’s use the gravity of the Moon. Drop the objects when the moon passes directly above the lower-altitude object. The Moon exerts a greater upward pull on the lower-altitude object, slowing its descent slightly more than it slows the descent of the higher-altitude object.
I don’t know. This Moon is a terrible idea, but it is real. The Moon is there.
A solution, I don’t exactly love, but what can I do.
Can you make it without the Moon?
You don’t seem to prefer aerodynamic solutions but they are abundant so here’s another one.
Orientation is unspecified so we can drop one object right-side-up and drop the other upside-down. A shape can be designed such that both orientations are stable (neither will switch orientations during free-fall) and such that one orientation has a different ballistic coefficient, creating a different terminal velocity. This could be accomplished with fan blades or pitot tubes of different sizes or protruding at different angles, converting different amounts of kinetic energy to heat depending on the direction of airflow.
The thing that brings my attention is the phrase “without any initial relative velocity against the planet”. They might not move relative to the planet, but Earth is not an inertial reference frame and is rotating, so bodies in different latitudes would have different speeds in relation to an inertial referential.
This might give some ideas to someone. But not very directly, I think.
My first thought was that B reaches terminal velocity and that’s it, but the object A is dropped from substantially higher altitude, picks up speed much higher than the terminal velocity and the atmosphere won’t slow it down enough,
I do not feel like oing the math, but there is a simpler solution: jnvg sbe n jnir naq gvzr gur qebc va fhpu n jnl gung bowrpg N, orvat uvture ol yrff guna gur jnir urvtug, uvgf gur perfg bs gur jnir.
There are no such special conditions for A and another special conditions for B. The sea and the air are roughly equal for both.
Well, if you said “the sea is rough equally for both of them” it would be obvious:-)
Another idea: A ebgngrf. Gur Zntahf rssrpg cebivqrf yvsg naq fybjf qbja gur snyy.
No, they both rotate or neither one rotates. They are equal, so the same rotation must be assumed. Their initial places are different and any difference only comes from that fact.
Still, an interesting suggestion.
If both rotate (I assume the same angular velocity) , what can be said about the direction of their axes of rotation?
Another idea: Gurl ner zntargvp, pybfr rabhtu gung gurl nggenpg gurzfryirf, fgvpx gbtrgure naq gur ubevmbagny nflzrgel pnhfrf gur bowrpgf gb ghzoyr naq gur uvture bar pbzrf qbja svefg.
This one does not need air-breaking. I like it!
But instead of magnetism, the gravity may work even better. Two massive objects with non-negligible gravity between themselves.
That’s my favorite idea.
If their gravity is significant enough, then it is incorrect to describe that they splash into the Atlantic—it’s the Atlantic that splashes into them.
I’d prefer solutions that do not destroy the Earth :-)
Two giant golden balls, dropped somewhere bellow the geosynchronous orbit might do the trick of a little orbiting around each other and then splashing into the ocean, one after another.
That might cause some damage, but the Earth would survive as a planet. Rather costly and not environment friendly solution.
What do you mean “costly”, we end up with two giant golden balls :-D
Or two ordinary small balls, one dropped from just above the geosynchronous orbit, the second one from far above the orbit. While the first one slowly drifts away to the space, the second shoots away, makes a complete (retrograde) orbit around Sun and splashes into the Atlantic while the first ball is still drifting...
Requires some careful timing, though.
This is true, but those Moon or Sun solutions aren’t my favorite. Moon, Sun, Jupiter and so on are external agents I’ve forgotten to explicitly forbid. Next time, I’ll be even more careful when posting a problem. :-)
Airplanes dropped in different orientations, or in a way that’s sensitive to initial conditions and leads to B gliding while A stalls. B is dropped from right above a passing eagle and gets carried off to Mordor. They’re “dropped” far out past geostationary orbit so that “stationary relative to the planet” in fact means that they’re flung off into space, and only reach Earth by getting slingshotted around other planets. Both are dropped over Brazil with notes to please throw them into the Atlantic, and A is dropped so that its coriolis motion as it falls will push it to a more visible area. They’re microscopic black holes dropped from the other side of the earth.
There is some wit here, but no proper solution.
Since you clearly have something in mind, once you reveal it, are we going to go “Oh, yeah, that’s much more sensible than the gliders that are dropped near the boundary of glide vs. stall air pressure,” or are we going to go “well, that’s arbitrary.”
Rigid hot-air-balloon shapes that start out at 1500 Celsius and fall to earth once they are no longer keeping the air under them hot. Seed crystals in a hailstorm. Any object that falls faster if broken and will break if dropped from the higher altitude. Solid-state electrostatic thrusters pointed downward, that arc and fail if the pressure is too high. Spinning propeller craft thrusting downward that undergo a laminar to turbulent transition if the pressure is too high.
As Lumifer said, “freefall” and as the initial conditions say—rigid bodies.
The problem specified “freefall”, so thrusters are out. But I agree that it’s underspecified—there are way too many things which fit so we are reduced to guessing what Thomas had in mind.
It is not what I have in mind. Anything goes, which does not break the initial conditions.
The initial conditions did not forbid dropping object A into a downdraft and object B into an updraft :-)
But here is another attempt: objects A and B open parachutes 3 seconds after passing altitude X. Object B starts at altitude X, accelerates from zero for three seconds, and then radically slows down. Object A starts higher, so when it passes altitude X it is already going fast and so in three seconds is capable of passing B which is already braked by a parachute.
Yes, well. There is some wit here again. The best solution I have in mind don’t require an atmosphere at all.
Well, maybe just for the sake of the liquid ocean water. Which is only for the sake of “a mountain and a valley soulution” prevention.
If this scenario is actually possible then it seems like the reason should be that the thing dropped from higher up gets to accelerate faster at first due to the thinner atmosphere. However, I’ve tried quite a lot of toy examples with different dependencies of air resistance on height and on speed, and none of them showed the trailing object ever quite catching up; unless I’ve screwed up or there’s some other thing going on (relating perhaps to the earth’s magnetic field or solar wind or something, but these seem like desperate longshots), either this doesn’t actually happen or it happens only when the circumstances are just right.
It should always happen, when there are no extreme conditions which would prevent it.