Do train gaps appear smaller in motion?

[Nabeshin]

A friend of mine posed this SR paradox to me a few weeks ago and I was unable to come up with a convincing answer (nor have I been convinced by any I've heard!). The problem is as follows:
Assume we have a train track which has certain gaps in it. At rest, the train which moves on the track is longer than the gaps (you can probably see where this is going already...). Now, when the train is in motion, from the train frame the track gaps are lorentz contracted. To the platform frame, the train is contracted and for a certain velocity, will actually be smaller than the gaps.

Now, it seems reasonable to me that the platform observer will see the train fall some distance. However, what troubles me is what the train observer will see. Owing to the fact that his train is very much larger than the gap in the tracks, intuition states he will notice no effect. In the (sparse) literature on this paradox, it seems that the train will indeed fall, but not much owing to it's high velocity. They somehow claim this solves the paradox, but I am more skeptical. The train cannot have some vertical motion in one frame and none in the other, for any discrepency will lead to a different outcome.

Feel free to posit the train as a rigid body, or perhaps use the more realistic assumption of a normal physical material. I feel the resolution to this lays somewhere in the mechanics of the train (in the trains frame) as it passes over the very slight gap, but cannot quite get things reconciled in my mind. Thoughts include some type of torquing motion where only part of the train is unsupported by the track.

Cheers

 

[starthaus]

A friend of mine posed this SR paradox to me a few weeks ago and I was unable to come up with a convincing answer (nor have I been convinced by any I've heard!). The problem is as follows:
Assume we have a train track which has certain gaps in it. At rest, the train which moves on the track is longer than the gaps (you can probably see where this is going already...). Now, when the train is in motion, from the train frame the track gaps are lorentz contracted. To the platform frame, the train is contracted and for a certain velocity, will actually be smaller than the gaps.

Now, it seems reasonable to me that the platform observer will see the train fall some distance. However, what troubles me is what the train observer will see. Owing to the fact that his train is very much larger than the gap in the tracks, intuition states he will notice no effect. In the (sparse) literature on this paradox, it seems that the train will indeed fall, but not much owing to it's high velocity. They somehow claim this solves the paradox, but I am more skeptical. The train cannot have some vertical motion in one frame and none in the other, for any discrepency will lead to a different outcome.

Feel free to posit the train as a rigid body, or perhaps use the more realistic assumption of a normal physical material. I feel the resolution to this lays somewhere in the mechanics of the train (in the trains frame) as it passes over the very slight gap, but cannot quite get things reconciled in my mind. Thoughts include some type of torquing motion where only part of the train is unsupported by the track.

Cheers

Rest assured that no matter how high the speed $v<c$, the train will not "fall through the cracks" .

 

[AJ Bentley]

The mistake is to assume that you can apply then Lorentz contraction in one dimension(x) whilst ignoring it in the perpendicular (y).
Of course that is going to lead to paradox.
A proper analysis of the movement of the train in both directions eliminates the problem.

 

[JesseM]

A friend of mine posed this SR paradox to me a few weeks ago and I was unable to come up with a convincing answer (nor have I been convinced by any I've heard!). The problem is as follows:
Assume we have a train track which has certain gaps in it. At rest, the train which moves on the track is longer than the gaps (you can probably see where this is going already...). Now, when the train is in motion, from the train frame the track gaps are lorentz contracted. To the platform frame, the train is contracted and for a certain velocity, will actually be smaller than the gaps.

Now, it seems reasonable to me that the platform observer will see the train fall some distance. However, what troubles me is what the train observer will see. Owing to the fact that his train is very much larger than the gap in the tracks, intuition states he will notice no effect. In the (sparse) literature on this paradox, it seems that the train will indeed fall, but not much owing to it's high velocity. They somehow claim this solves the paradox, but I am more skeptical. The train cannot have some vertical motion in one frame and none in the other, for any discrepency will lead to a different outcome.

Feel free to posit the train as a rigid body, or perhaps use the more realistic assumption of a normal physical material. I feel the resolution to this lays somewhere in the mechanics of the train (in the trains frame) as it passes over the very slight gap, but cannot quite get things reconciled in my mind. Thoughts include some type of torquing motion where only part of the train is unsupported by the track.

Cheers

Even if the gap is smaller than the train, won't the front of the train start tipping downwards when it reaches the gap (and when the very front has not yet reached the position where the gap ends and the tracks start again)? A detailed analysis would presumably require modeling both the normal force upward from the tracks and the "gravitational" force downward (you can't actually have gravity in special relativity, but we could consider an electromagnetically charged train with an oppositely-charged planet below, or perhaps assume that the tracks were mounted in a rocket which was continuously accelerating upward)

 

[starthaus]

Even if the gap is smaller than the train, won't the front of the train start tipping downwards when it reaches the gap (and when the very front has not yet reached the position where the gap ends and the tracks start again)?

Since the gap is always much smaller than the diameter of the wheels, this doesn't happen. The wheels always make contact with either one or with two adjoining tracks. One way to convince yourself is that , in the frame of the train, the wheels always ride on the tracks, even when they go across the gaps. (otherwise you would have a very rough ride :-)). So, in the frame of the train, the normal reaction of the track against the wheels is always non-zero.

 

[starthaus]

The mistake is to assume that you can apply then Lorentz contraction in one dimension(x) whilst ignoring it in the perpendicular (y).
Of course that is going to lead to paradox.
A proper analysis of the movement of the train in both directions eliminates the problem.

There is no length contraction in the direction transverse to the direction of motion.

 

[JesseM]

Since the gap is always much smaller than the diameter of the wheels, this doesn't happen. The wheels always make contact with either one or with two adjoining tracks.

I don't understand, isn't a single wheel only in contact with one track? How can it be in contact with "adjoining tracks"?

One way to convince yourself is that , in the frame of the train, the wheels always ride on the tracks, even when they go across the gaps. (otherwise you would have a very rough ride :-)).

Are there "gaps" in ordinary train tracks? And if so, are you claiming that when the wheels pass over them, they don't experience any downward falling motion, even if they only have time to fall a fraction of a millimeter in the time they pass over the gap?

And if you make the gap "much smaller than the diameter of the wheels" in the train frame, but make the train move so close to light speed that the gap is larger than the entire length of the train in the platform frame, then surely the train would be moving so fast that it wouldn't have time to fall very far before reaching the other end of the gap; I trust that a full calculation would show it only fell by the same fraction of a millimeter (or whatever) as each wheel briefly fell when crossing the gap in the train's own frame. Like I said, you would need to do a detailed calculation in which the train was a non-rigid object which could flex and bend, and where you'd take into account the normal force from every point on the tracks in contact with the wheels, along with whatever force was standing in for gravity...but as long as all the laws involved are Lorentz-invariant ones, it's guaranteed that both frames should agree on the maximum vertical displacement experienced by each wheel in crossing the gap.

 

[starthaus]

I don't understand, isn't a single wheel only in contact with one track? How can it be in contact with "adjoining tracks"?

When the wheel goes ove the gap between two adjoining tracks it makes contact with both tracks at the same time.

Are there "gaps" in ordinary train tracks?

Not "in". "Between" adjoining sections. This is what the OP is about.

And if so, are you claiming that when the wheels pass over them, they don't experience any downward falling motion, even if they only have time to fall a fraction of a millimeter in the time they pass over the gap?

Yes, the wheels will go down by a fraction of mm. How is this relevant to your claim that the car will tip between the gaps? In the frame of the train, the wheels are in contact with the rails at all times

And if you make the gap "much smaller than the diameter of the wheels" in the train frame, but make the train move so close to light speed that the gap is larger than the entire length of the train in the platform frame, then surely the train would be moving so fast that it wouldn't have time to fall very far before reaching the other end of the gap; I trust that a full calculation would show it only fell by the same fraction of a millimeter (or whatever) as each wheel briefly fell when crossing the gap in the train's own frame.

The point is that the train will not fall at all since the normals on the wheels is non-null.

Like I said, you would need to do a detailed calculation in which the train was a non-rigid object which could flex and bend, and where you'd take into account the normal force from every point on the tracks in contact with the wheels, along with whatever force was standing in for gravity...but as long as all the laws involved are Lorentz-invariant ones, it's guaranteed that both frames should agree on the maximum vertical displacement experienced by each wheel in crossing the gap.

You don't need that. You know that the normal reaction on the wheels is non-null in either the train frame (this is obvious), neither in the track frame (this is derived from analyzing the normal forces to the wheels). In other words: the wheels rest on the rails in both the train frame and in the track frame. The train doesn't fall thruogh the gaps in the train frame, neither does it fall in the track frame. This completely resolves the false "paradox"

 

[JesseM]

Yes, the wheels will go down by a fraction of mm. How is this relevant to your claim that the car will tip between the gaps? In the frame of the train, the wheels are in contact with the rails at all times

My claim was originally based on the assumption the gap was substantial in both frames, the OP didn't say anything about the gap being much smaller than the diameter of the wheel, that seems to be a new condition that you are introducing. In any case I think my statement is still correct, since if the frontmost wheels dips down by a fraction of a mm, then the front of the train will tip down slightly. "Tip" did not imply that the front of the train would fail to continue to ride the tracks after crossing the gap.

The point is that the train will not fall at all since the normals on the wheels is non-null.

If the wheels dip down by a fraction of a mm, surely the train car connected to the wheels does as well? I didn't mean "fall" to imply free-fall, any small reduction in the normal force upwards resulting in net downward acceleration would count, even if brief and soon countered by an upward acceleration.

 

[starthaus]

My claim was originally based on the assumption the gap was substantial in both frames, the OP didn't say anything about the gap being much smaller than the diameter of the wheel, that seems to be a new condition that you are introducing.

No, I am not introducing it, it is a fact of life.

In any case I think my statement is still correct, since if the frontmost wheels dips down by a fraction of a mm, then the front of the train will tip down slightly. "Tip" did not imply that the front of the train would fail to continue to ride the tracks after crossing the gap.

Irrelevant, there is no way the train can pass through the gap since the normal reaction to the wheel is non-null in both frames at all times.

If the wheels dip down by a fraction of a mm, surely the train car connected to the wheels does as well? I didn't mean "fall" to imply free-fall, any small reduction in the normal force upwards resulting in net downward acceleration would count, even if brief and soon countered by an upward acceleration.

True but irrelevant. The train can't go throgh the gap. The non-null reaction of the track prevents it from doing that. This is not a kinematics problem, this is a static mechanics problem.

 

[yossell]

No, I am not introducing it, it is a fact of life.

Given our slow moving trains, yes. But I think the point of the paradox is that, if there is any gap at all then, from the stationary person's point of view, since there is no positive lower bound on the degree to which something can be lorentz contracted, if the train reaches a great enough speed its wheels (and even the train itself) will be (relative to the stationary observer) narrower than the gap - just by Lorentz contraction.

Accordingly, for a very short while, the train is unsupported in the observer's frame.

 

[starthaus]

Given our slow moving trains, yes. But I think the point of the paradox is that, if there is any gap at all then, from the stationary person's point of view, since there is no positive lower bound on the degree to which something can be lorentz contracted, if the train reaches a great enough speed its wheels (and even the train itself) will be (relative to the stationary observer) narrower than the gap - just by Lorentz contraction.

Accordingly, for a very short while, the train is unsupported in the observer's frame.

This is false. You are trying to solve this problem as a kinematics problem (which it isn't).

 

[curiousphoton]

Guys,

Isn't this thought experiment just a version of Einstein's original though experiment that led him to SR? You know the one where he visualized someone dropping a rock out of a window of the train? To the person in the train, the rocks path is curved. To a person watching the train go by, the rocks path is straight down. Both make different observations / conclusions so who is right? According to Einstein, they BOTH are.

Back to the current thought experiment. The train will experience some downward displacement...the amount depending on your RF.

 

[JesseM]

No, I am not introducing it, it is a fact of life.

It's a thought-experiment, not life--in real life it's not practically possible for a train to travel at relativistic speeds! If one wants to consider a thought experiment where a large section of the track has been removed so the gap is half the size of the train, that's a perfectly legitimate thought-experiment which relativity should be able to deal with too.

Irrelevant, there is no way the train can pass through the gap since the normal reaction to the wheel is non-null in both frames at all times.

Merely saying it's "non-null" is not sufficient to show both frames make precisely the same predictions, and it does not address the thought-experiment where the gap is large enough that a wheel can lose all contact with the tracks for some period of time, which may well have been what the OP was imagining.

True but irrelevant. The train can't go throgh the gap. The non-null reaction of the track prevents it from doing that. This is not a kinematics problem, this is a static mechanics problem.

Again "non-null" is too vague, if the normal force upward is less than the force downwards the wheel will accelerate downwards, if the wheel was sufficiently flexible it might squeeze through the gap like a glob of pudding, but more realistically once different points on the wheel are in contact with either side of the track at the front and back of the gap, the normal force will increase to balance out the downward force so the wheel can't go down any further.

 

[starthaus]

Guys,

Isn't this thought experiment just a version of Einstein's original though experiment that led him to SR? You know the one where he visualized someone dropping a rock out of a window of the train? To the person in the train, the rocks path is curved. To a person watching the train go by, the rocks path is straight down. Both make different observations / conclusions so who is right? According to Einstein, they BOTH are.

Back to the current thought experiment. The train will experience some downward displacement...the amount depending on your RF.

In the frame of the train the train does not exhibit any "downward displacement" let alone fall through the gap between rails. Relativity forbids a different outcome of this experiment in the frame of the track (or any other inertial frame), i.e. the train does not fall through the gap no matter how large its speed wrt the tracks.

 

[Vanadium 50]

This is a relativity of simultaneity problem:

Train frame order of events:

1. Front wheels enter gap
2. Front end of train begins to dip
3. Front wheels catch the tracks on the far side
4. Front end of train straightens out
5. Rear wheels enter gap
6. Rear end of train begins to dip
7. Rear wheels catch the tracks on the far side
8. Rear end of train straightens out

Station frame order of events:

1. Front wheels enter gap
2. Front end of train begins to dip
3. Rear wheels enter gap
4. Rear end of train begins to dip
5. Front wheels catch the tracks on the far side
6. Front end of train straightens out
7. Rear wheels catch the tracks on the far side
8. Rear end of train straightens out

 

[M Grandin]

Maybe a simple answer is that SR is about constant velocities without accelerations.
But passing a gap, falling down somewhat due to gravity, implies acceleration - and therefore no longer SR applies. GR may solve the "paradox".

 

[Al68]

Since the gap is always much smaller than the diameter of the wheels, this doesn't happen.

LOL. It seems obvious that the intent of the OP was that the gap is large enough so that the scenario is meaningful as an SR thought experiment.

Why don't we just assume a train rest length of 100 meters, a gap of 80 meters rest length, and v=0.8c. I think that would meet the intent of the OP.

This is a relativity of simultaneity problem...

Yes, this is just a variation of the barn/pole paradox. The only thing the frames will disagree on is when each part of the train "dropped".

 

[yossell]

Accordingly, for a very short while, the train is unsupported in the observer's frame.

This is false. You are trying to solve this problem as a kinematics problem (which it isn't).

I respectfully disagree on two counts. Firstly, I'm not trying to solve it, I'm just trying to present the problem.

Secondly, the formula for length contraction is: $$L' = L\sqrt{1-\frac{v^2}{c^2}}$$
It's clear that, by a high enough v, you can make L' as small as you like. So, if there's some space c between the tracks, then there's some v such that the train is contracted to a length smaller than c. So, for a period of time, it's unsupported.

Vanadium's points about relativity of simultaneity explain why you can't argue from the fact that there's always contact in train's frame to there always being contact in stationary frame.

edit: Beaten again. Jeez

 

[JesseM]

Maybe a simple answer is that SR is about constant velocities without accelerations.
But passing a gap, falling down somewhat due to gravity, implies acceleration - and therefore no longer SR applies. GR may solve the "paradox".

Not true, SR can deal with accelerations fine, it's only gravity and curved spacetime that requires GR. You can analyze the motion of an accelerating object from the perspective of an inertial frame, and even if you choose to use an accelerating frame, the modern perspective is that this is still part of "SR" as long as there is no spacetime curvature. See this section of the Usenet Physics FAQ for more info.

 

[starthaus]

LOL. It seems obvious that the intent of the OP was that the gap is large enough so that the scenario is meaningful as an SR thought experiment.

You obviously never seen a railway track.

Why don't we just assume a train rest length of 100 meters, a gap of 80 meters rest length,

LOL.

 

[JesseM]

You obviously never seen a railway track.

Have you ever seen a train moving at relativistic speed? Are you unfamiliar with the term "thought-experiment"?

 

[curiousphoton]

Have you ever seen a train moving at relativistic speed?

Yes, BART on a Friday afternoon

 

[M Grandin]

Not true, SR can deal with accelerations fine, it's only gravity and curved spacetime that requires GR. You can analyze the motion of an accelerating object from the perspective of an inertial frame, and even if you choose to use an accelerating frame, the modern perspective is that this is still part of "SR" as long as there is no spacetime curvature. See this section of the Usenet Physics FAQ for more info.

Thanks for comments. I also once learned that SR in some respects admits accelerations.
SR could be used when integrating total elapsed time, passed distance etc. for accelerated rockets etc.

But at least in this case it is an acceleration due to gravity ("curved space"), why (according to yourself) GR should be required - but in some sense I must have misunderstood you.

 

[JesseM]

But at least in this case it is an acceleration due to gravity ("curved space"), why (according to yourself) GR should be required - but in some sense I must have misunderstood you.

Well, nothing about the problem requires that it be true gravity. As I said in post #4 on this thread:

A detailed analysis would presumably require modeling both the normal force upward from the tracks and the "gravitational" force downward (you can't actually have gravity in special relativity, but we could consider an electromagnetically charged train with an oppositely-charged planet below, or perhaps assume that the tracks were mounted in a rocket which was continuously accelerating upward)

 

[starthaus]

This is a relativity of simultaneity problem:

Train frame order of events:

[*]Front wheels enter gap

[*]Front end of train begins to dip

The above is unphysical since the gap between tracks is 1-2mm for every 20m of track.


The train cannot fall through the gap in the train frame, nor can it do that for any speed (relativistic or not) in the track frame.

 

[starthaus]

Have you ever seen a train moving at relativistic speed? Are you unfamiliar with the term "thought-experiment"?

No, I am not. By the same token, trains going over gaps of the length of a car (or locomotive) are unphysical. I understand your point, please make the effort to understand mine.

 

[JesseM]

The above is unphysical since the gap between tracks is 1-2mm for every 20m of track.

No one is talking about real trains or real train-tracks, it's a thought-experiment about relativity. If you keep talking about the size of actual gaps in actual train tracks without addressing the fact that this is an imaginary thought-experiment (in which a train is moving at relativistic speed), then the obvious conclusion is going to be that you're just trolling.

 

[starthaus]

No one is talking about real trains or real train-tracks, it's a thought-experiment about relativity. If you keep talking about the size of actual gaps in actual train tracks

This is precisely what I specified from the first post. If you want to talk about flying trains, be my guest.

without addressing the fact that this is an imaginary thought-experiment (in which a train is moving at relativistic speed), then the obvious conclusion is going to be that you're just trolling.

Why don't you ask the OP what he's talking about?

 

[JesseM]

No, I am not. By the same token, trains going over gaps of the length of a car (or locomotive) are unphysical.

"Unphysical" means forbidden by the laws of physics, not just difficult in a practical sense. If you don't think a train moving at a substantial fraction of light speed is "unphysical", then how can it be "unphysical" to create a large gap in a train track? (you could do it by blowing up the center of a bridge, for example) Or are you just saying it's "unphysical" that a train could actually make it over such a large gap as opposed to falling down? If so, note that the original post didn't specify whether the train fell or made it over the gap, the problem was just to show how both frames would agree on what happens. Anyway, if the gap is 10 meters in the platform train but the train is traveling at 0.999c, each wheel would only take 33 nanoseconds to get from one end to the other, not enough time to fall very far...so, if you accept the existence of relativistic trains in the first place, I don't see why you couldn't accept that the wheel will connect smoothly with the track on the far side of a large gap.

This is precisely what I specified from the first post. If you want to talk about flying trains, be my guest

This is a thread for discussing the scenario described in the OP, you don't get to put arbitrary restrictions on the scenario and then expect everyone else to discuss your own restricted scenario rather than the more general one in the OP. And like I said, the OP doesn't specify whether the train makes it or not, so if your sarcastic "flying trains" comment implies you think it would automatically fall through a large gap (and that this would be predicted in both frames), feel free to try to defend that position.

 

[starthaus]

"Unphysical" means forbidden by the laws of physics, not just difficult in a practical sense. If you don't think a train moving at a substantial fraction of light speed is "unphysical", then how can it be "unphysical" to create a large gap in a train track? (you could do it by blowing up the center of a bridge, for example) Or are you just saying it's "unphysical" that a train could actually make it over such a large gap as opposed to falling down?

Really easy, maglev trains move at very high speeds. The gaps in their tracks are 1mm. It is easy to imagine a 1km straight track for a maglev moving at very high speed.
By the same token , a train trying to make it over a 10m gap will certainly derail.

if the gap is 10 meters but the train is traveling at 0.5c, each wheel would only take 33 nanoseconds to get from one end to the other, not enough time to fall very far...so, if you accept the existence of relativistic trains in the first place, I don't see why you couldn't accept that the wheel will connect smoothly with the track on the far side of a large gap.

Err,as I was saying, if the train were moving at 50km/h it will for sure derail. So, this is not a very interesting problem.
I explained the boundary conditions for the problem I solved in post 3, you never acknowledged that the solution is correct. Can you find it in yourself to do so?

 

[JesseM]

Really easy, maglev trains move at very high speeds.

Not at relativistic speeds great enough that in the platform frame the train's length contraction will make it shorter than the gaps! Did you even read the OP?

The problem is as follows:
Assume we have a train track which has certain gaps in it. At rest, the train which moves on the track is longer than the gaps (you can probably see where this is going already...). Now, when the train is in motion, from the train frame the track gaps are lorentz contracted. To the platform frame, the train is contracted and for a certain velocity, will actually be smaller than the gaps.

I explained the boundary conditions for the problem I solved in post 3

This isn't a thread about your problem, it's a thread about the problem in the OP. Feel free to start a thread about your own problem if you find it interesting, although it has nothing to do with relativity so it wouldn't belong in this forum.

 

[starthaus]

Not at relativistic speeds great enough that in the platform frame the train's length contraction will make it shorter than the gaps! Did you even read the OP?

Yes, I read the OP. I have also had enough exchanges with you to see that you never understood my solution.

This isn't a thread about your problem, it's a thread about the problem in the OP.

The OP has specified "gaps". He never came back to specify the size of those gaps. I elected to use the real life size. You may elect to use the size jumped by Batman trains. <shrug>

Feel free to start a thread about your own problem if you find it interesting, although it has nothing to do with relativity so it wouldn't belong in this forum.

...but, IF the gaps are big, then the train will fall through them at low speeds. So, the problem is not relativstic in this case, therefore it does not belong in this forum.

 

[JesseM]

The OP has specified "gaps". He never came back to specify the size of those gaps. I elected to use the real life size. You may elect to use the size jumped by Batman trains. <shrug>

He said that in the platform frame, the speed of the train was so great that it was length-contracted down to a size smaller than the gaps. So if you want the gaps to be 1 mm, that's fine, but then if the train is a mere 10 meters long it must have a speed which gives it a length contraction factor greater than 10,000, implying it's moving at just a hair under the speed of light (do you actually know what 'length contraction' is in relativity, and how to calculate it?)

If you continue to talk about small gaps and trains with speeds too small for the length contraction factor to make their length shorter than the gaps, I'll conclude you're definitely trolling and report your posts to the moderators.

 

[starthaus]

He said that in the platform frame, the speed of the train was so great that it was length-contracted down to a size smaller than the gaps. So if you want the gaps to be 1 mm, that's fine, but then if the train is a mere 10 meters long it must have a speed which gives it a length contraction factor greater than 10,000, implying it's moving at just a hair under the speed of light (do you actually know what 'length contraction' is in relativity, and how to calculate it?)

Yes, I do very well. The point that you still fail to understand is that no matter how high the speed , if the gap is smaller than the diameter of the wheels, the train will never fall through the gaps.

If you continue to talk about small gaps and trains with speeds too small for the length contraction factor to make their length shorter than the gaps, I'll conclude you're definitely trolling and report your posts to the moderators.

Threats are not a form of scientific argument. Especially coming from a "science advisor". You can easily imagine a gap of 50cm. If the wheel diameter is 60cm then the train will not fall through the gap, no matter what fraction of "c" it is going. I explained that to you earlier, I will explain it one last time: the normal reaction from the track is not zero when the train s at rest wrt the track. Give the train the speed 0.9999c and the normal transforms into a non-zero value. Therfore, the train never leaves the track.

You are trying to reduce the problem to the well known problem of the "manhole and the cover" (Vanadium gave a very good description of that problem , as seen in Taylor and Wheeler's "Spacetime Physics"). The context is different for that problem, the manhole is large nough to allow the cover to tilt and fall through even at low speeds.