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The Impossible Physics Problem

Roboramma said:
All of the boxes could be drifting, but in different currents. But they'd still be drifting (or not) relative to the center of mass of the ocean as a whole.
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Well, yes, but the original problem didn't say to determine the movement relative to the center of mass of the ocean as a whole. You might have thought that was clearly implied by "how fast your box is drifting through the ocean" but to me it doesn't imply that at all. When you're in the ocean what matters is the water immediately around you. "How fast can you swim through the ocean?" can be answered without regard for what current you might be in, unlike e.g. "How fast can you swim towards shore?" Similarly if you were to ask how fast a bird is flying through the air, I'd assume you meant the air where the bird is, not the ground or the average wind of the entire continent.

Here's a simpler model for the intended point: "You're riding in a bus along a long straight level highway. Some distance ahead of you is a truck that your bus, and some distance behind you is a sports car. The distance from your bus to the truck is decreasing at a rate of 1 km per hour, and the distance from your bus to the sports car is decreasing at a rate of 6 km per hour. What's the bus driver's name? How fast is the bus going?
 
PhantomWolf said:
You awake to find yourself in a box. Inside the box with you are a stopwatch, a compass, and a laser range finder.
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I've got a bad feeling about this.

PhantomWolf said:
Using your compass you are able to determine that the box is orientated so that the walls are to your north, south, east, and west. Using the stopwatch and rangefinder you are about to determine your change in distance to each of the other boxes over the course of a minute. These are as follows. Multiple checks show that this change is consistent.

The north box -3m
The south box +2m
The west box +4m
The east box +1m

Using the above information, determine how fast your box is drifting through the ocean, and in which direction.
Click to expand...
I don't think I can, but I do believe that, under some special circumstances, I may be able to estimate the rate of my drift and also obtain some partial information about the direction.

The rangefinder is superfluous, unless it contains a bit of ferrous metal I could obtain by taking it apart. The relevant tools are the stopwatch, magnetic compass, the slits, my knowledge of the earth's dimensions, and my superhuman ability to detect and to quantify even the slightest accelerations.

Thanks to my superhuman ability to detect acceleration, I can detect any changes to the direction or rate of my drift and correct for them. I can also detect and correct for any spinning of my coffin box. Without loss of generality, therefore, I will describe a special case and a more general case without taking into account the possibility that the direction and rate of my drift varies with time.

If I'm drifting directly toward magnetic north, and I keep my compass parallel to the ocean's surface (as seen through the slits) but otherwise in a fixed orientation relative to the geodesic (as confirmed by the accelerometer), I will eventually pass over magnetic north, causing my compass first to wander and then to swing around. At that point I'm drifting toward magnetic south, which I will eventually pass over. Using the stopwatch to time how long it takes to go from magnetic north to magnetic south tells me my rate of drift.

If that happens, I know when I'm drifting toward magnetic north, and I know when I'm drifting toward magnetic south.

If that doesn't happen, I know I'm not drifting along the geodesic between magnetic north and south. That's not much, but it does give me a tiny bit of information about my drift.

With a superhuman accelerometer, or a bit of ferrous metal, I could use the compass to detect changes in the strength of the magnetic field, which would give me more partial information about my direction and rate of drift.

ETA: I just noticed that sphenisc found a better use for the rangefinder.
 
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The only relative motion thing that I always forget how it fits in is rotation. The old how does it know if it’s spinning or not thing. My Google fu always fails when I realize I’ve forgotten it again. It’s worse than when I forget whether or not Nicholas Cage was in something at the same time as I forget his name is Nicholas Cage.
 
This guy’s description makes the most sense to me, don’t know enough about it to know how good it is though. Alec Zander, nom de plume….

Linear motion is often described as being relative to the frame of reference of other bodies in flat space-time.

Rotation is actually relative to the frame of reference of the potential energy function (in practice, a volume of space) within which the bodies are rotating.

Bodies in translation are following the effective potential energy surface of an unconstrained (or weakly constrained) system which corresponds to moving over a mostly flat effective potential energy surface (and in most examples, mostly through flat space-time).

Bodies in rotation are following the effective potential energy surface of a strongly constrained system, where a deep (effectively inescapable) potential energy minimum is defined along a vector between the rotating masses: the radial vector (also called N, the normal unit vector, in the Frenet–Serret formulas - frame).

There is confusion on this topic because of the conflation of:

The movement of bodies in flat or warped space-time

and

The movement of bodies over a potential energy surface in phase space

which are similar, but not the same, since it is possible to have both effects at the same time (for example, movement of bodies constrained by an electromagnetic field in a gravitationally flat space). In other words, while the space-time the objects are rotating in may be flat, the potential energy surface of the rotating system (over which they are moving) IS NOT FLAT, and has a natural frame of reference defined by the axial and equatorial anisotropy of its own motion.
The spatial orientation of the potential energy function of the rotating system can only be defined with respect to the components of the rotating system (internal coordinates).

The potential energy volume (as a higher dimensional analog of a potential energy surface) of a rotating system is not just anisotropic, it has specifically favored axes, the first of which is the radial vector of rotation - because, by physical definition of the system, it is constant.
The very physical constraint (whether it is a rope between buckets or gravity between celestial bodies) that causes rotation is therefore the spatial axis along which the effective potential energy function is NOT FLAT, and that's why it is the basis of the natural frame of reference for rotating systems. The natural frame of reference of the potential energy volume is the potential energy parameter which is constant - the radial vector.

Because it has both a fixed length and a fixed angular relationship to the other force vector, the radial vector is the natural basis of the frame of reference.

The radial vector has two invariant relationships:

In steadily rotating systems, by definition, the radial vector's length is fixed (Frenet frame vector N, the normal unit vector).

In addition, the non-radial vectors (the Frenet frame tangent vector T, and the vector orthogonal to both N and T, called B for binormal) have fixed angular relationships (90 degrees) to N (aka the radial vector).

Together, the Frenet frame for a rotating body reveals the main axis of the rotational system's potential energy function - that invisible thing we forget to visualize! All force evaluations upon rotating bodies must be computed along this physical axis in order to have relevance to the physical potential energy function.

Centripetal force arises because at every instant the tangent and binormal vectors have components which point away from the radial restraint vector. Summed and averaged over a complete rotation, the net vector is precisely outward, in opposition to the inward radial constraint.
 
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PhantomWolf said:
And that's kinda the point. That all velocities and thus all movement is relative, and never absolute. Hence why it is a question to give people that can't understand that movement is relative.
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No. It's a very bad question to give to such people. It is far too complex a scenario. It does not distill the essence of that concept into something easy to understand. People who do not understand the relativity of movement will not be enlightened by the problem, they will just get confused.
 
W.D.Clinger said:
I've got a bad feeling about this.


I don't think I can, but I do believe that, under some special circumstances, I may be able to estimate the rate of my drift and also obtain some partial information about the direction.

The rangefinder is superfluous, unless it contains a bit of ferrous metal I could obtain by taking it apart. The relevant tools are the stopwatch, magnetic compass, the slits, my knowledge of the earth's dimensions, and my superhuman ability to detect and to quantify even the slightest accelerations.

Thanks to my superhuman ability to detect acceleration, I can detect any changes to the direction or rate of my drift and correct for them. I can also detect and correct for any spinning of my coffin box. Without loss of generality, therefore, I will describe a special case and a more general case without taking into account the possibility that the direction and rate of my drift varies with time.

If I'm drifting directly toward magnetic north, and I keep my compass parallel to the ocean's surface (as seen through the slits) but otherwise in a fixed orientation relative to the geodesic (as confirmed by the accelerometer), I will eventually pass over magnetic north, causing my compass first to wander and then to swing around. At that point I'm drifting toward magnetic south, which I will eventually pass over. Using the stopwatch to time how long it takes to go from magnetic north to magnetic south tells me my rate of drift.

If that happens, I know when I'm drifting toward magnetic north, and I know when I'm drifting toward magnetic south.

If that doesn't happen, I know I'm not drifting along the geodesic between magnetic north and south. That's not much, but it does give me a tiny bit of information about my drift.

With a superhuman accelerometer, or a bit of ferrous metal, I could use the compass to detect changes in the strength of the magnetic field, which would give me more partial information about my direction and rate of drift.

ETA: I just noticed that sphenisc found a better use for the rangefinder.
Click to expand...


Not bad!

I was going to reassemble the instruments, along with pieces of the packaging of my food and water supply (in the absence of which, the question would be moot, as I pointed out before), to make a sensitive pressure gage. When placed at each of the four slits, that could tell me if wind was acting upon the box, or if an engine were propelling the box. (But not which one or in what combination, and if the engine were propelling the box exactly with the wind, it would fail to detect any variation at all.)

I decided to reassemble the instruments into a fully stocked bar and a video-on-demand system instead.
 
God knows!
You better start praying, if your in a box adrift in the middle of the ocean!
(It's a fine thought experiment, but God trumps thoughts.)

Uh oh! There are Skeptics in the other boxes. No Sunday School for you! :wackyunsure::wackywink:
 
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theprestige said:
I thought it mattered to every problem of motion.
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There is one nitpick or cheat in this case, if you play games with the wording of the question. You could say that in a reference frame where your box is standing still the answer is zero. But that doesn't meet the spirit of the question, it just means you should then tell us how fast the ocean is moving past you and in what direction.

Otherwise, choice of reference frame is arbitrary. Choose whichever frame makes the problem simplest for you. And state your chosen reference frame in the answer.
 
Myriad said:
Not bad!

I was going to reassemble the instruments, along with pieces of the packaging of my food and water supply (in the absence of which, the question would be moot, as I pointed out before), to make a sensitive pressure gage. When placed at each of the four slits, that could tell me if wind was acting upon the box, or if an engine were propelling the box. (But not which one or in what combination, and if the engine were propelling the box exactly with the wind, it would fail to detect any variation at all.)

I decided to reassemble the instruments into a fully stocked bar and a video-on-demand system instead.
Click to expand...

I salute your far more practical solution to the problem.
 
RecoveringYuppy said:
There is one nitpick or cheat in this case, if you play games with the wording of the question. You could say that in a reference frame where your box is standing still the answer is zero. But that doesn't meet the spirit of the question, it just means you should then tell us how fast the ocean is moving past you and in what direction.

Otherwise, choice of reference frame is arbitrary. Choose whichever frame makes the problem simplest for you. And state your chosen reference frame in the answer.
Click to expand...

I'm not all that enthusiastic about the spirit of the question either. I don't think this kind of Socratic approach actually works in the real world.
 
Myriad said:
Well, yes, but the original problem didn't say to determine the movement relative to the center of mass of the ocean as a whole. You might have thought that was clearly implied by "how fast your box is drifting through the ocean" but to me it doesn't imply that at all.
Click to expand...

Oh, I didn't think it implied that either. I was just responding to your statement that it's not drifting. If I'm standing somewhere (on a boat) or swimming near a box, and it starts to move away, I'll say to my friend "the box is drifting away", even though it' may not be moving relative to the local water around it.

Of course, you did qualify your statement by saying that it may be drifting relative to the ocean as whole, or to other parts of it, so whatever point I'm trying to make, you clearly already got it.

And your point that the box, if it's drifting, isn't moving with respect to the water around it, is probably a meaningful one.
 
Lithrael said:
The only relative motion thing that I always forget how it fits in is rotation. The old how does it know if it’s spinning or not thing. My Google fu always fails when I realize I’ve forgotten it again. It’s worse than when I forget whether or not Nicholas Cage was in something at the same time as I forget his name is Nicholas Cage.
Click to expand...
Anything that is rotating* is actually a collection of things trying to go in straight lines but being turned by forces between them and other objects.

So everything in a rotating object is undergoing acceleration, so you wouldn't expect the whole thing to be as simple as as linear unaccelerated motion.

If you have a rotating frame of reference and you have something with no forces acting on it, it won't go in a straight line, it will follow a curved path.

So if you have an object in space and a robot with a camera floating right at the centre and you can see the interior of the space station spinning around the camera and are wondering whether the robot is spinning inside the space station or the space station is spinning around the robot, just get the robot to pick up an object and release it in front of the camera. If the object stays put then the space station is spinning. If the object starts to move away in a curved path the robot is spinning.

* of course I am not referring to quantum spin, which is not a rotation.
 
Robin said:
Anything that is rotating* is actually a collection of things trying to go in straight lines but being turned by forces between them and other objects.

So everything in a rotating object is undergoing acceleration, so you wouldn't expect the whole thing to be as simple as as linear unaccelerated motion.

If you have a rotating frame of reference and you have something with no forces acting on it, it won't go in a straight line, it will follow a curved path.

So if you have an object in space and a robot with a camera floating right at the centre and you can see the interior of the space station spinning around the camera and are wondering whether the robot is spinning inside the space station or the space station is spinning around the robot, just get the robot to pick up an object and release it in front of the camera. If the object stays put then the space station is spinning. If the object starts to move away in a curved path the robot is spinning.

* of course I am not referring to quantum spin, which is not a rotation.
Click to expand...

So the "Am I rotating or is that thing spinning around me?" question just comes down to "What path would I see a body following if it had no forces acting on it?"

Or possibly, "What path would I see light taking?"
 
Lithrael said:
This guy’s description makes the most sense to me, don’t know enough about it to know how good it is though. Alec Zander, nom de plume….

Linear motion is often described as being relative to the frame of reference of other bodies in flat space-time.

Rotation is actually relative to the frame of reference of the potential energy function (in practice, a volume of space) within which the bodies are rotating.

Bodies in translation are following the effective potential energy surface of an unconstrained (or weakly constrained) system which corresponds to moving over a mostly flat effective potential energy surface (and in most examples, mostly through flat space-time).

Bodies in rotation are following the effective potential energy surface of a strongly constrained system, where a deep (effectively inescapable) potential energy minimum is defined along a vector between the rotating masses: the radial vector (also called N, the normal unit vector, in the Frenet–Serret formulas - frame).

There is confusion on this topic because of the conflation of:

The movement of bodies in flat or warped space-time

and

The movement of bodies over a potential energy surface in phase space

which are similar, but not the same, since it is possible to have both effects at the same time (for example, movement of bodies constrained by an electromagnetic field in a gravitationally flat space). In other words, while the space-time the objects are rotating in may be flat, the potential energy surface of the rotating system (over which they are moving) IS NOT FLAT, and has a natural frame of reference defined by the axial and equatorial anisotropy of its own motion.
The spatial orientation of the potential energy function of the rotating system can only be defined with respect to the components of the rotating system (internal coordinates).

The potential energy volume (as a higher dimensional analog of a potential energy surface) of a rotating system is not just anisotropic, it has specifically favored axes, the first of which is the radial vector of rotation - because, by physical definition of the system, it is constant.
The very physical constraint (whether it is a rope between buckets or gravity between celestial bodies) that causes rotation is therefore the spatial axis along which the effective potential energy function is NOT FLAT, and that's why it is the basis of the natural frame of reference for rotating systems. The natural frame of reference of the potential energy volume is the potential energy parameter which is constant - the radial vector.

Because it has both a fixed length and a fixed angular relationship to the other force vector, the radial vector is the natural basis of the frame of reference.

The radial vector has two invariant relationships:

In steadily rotating systems, by definition, the radial vector's length is fixed (Frenet frame vector N, the normal unit vector).

In addition, the non-radial vectors (the Frenet frame tangent vector T, and the vector orthogonal to both N and T, called B for binormal) have fixed angular relationships (90 degrees) to N (aka the radial vector).

Together, the Frenet frame for a rotating body reveals the main axis of the rotational system's potential energy function - that invisible thing we forget to visualize! All force evaluations upon rotating bodies must be computed along this physical axis in order to have relevance to the physical potential energy function.

Centripetal force arises because at every instant the tangent and binormal vectors have components which point away from the radial restraint vector. Summed and averaged over a complete rotation, the net vector is precisely outward, in opposition to the inward radial constraint.
Click to expand...

I don't understand a word of what he is saying, but even if it is right it is certainly much too complicated for a lay understanding this at a classical level.
 
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