2m car near lightspeed fall into 1m hole?

[jester1989]

will a 2m(rest length) car moving near lightspeed fall in a 1m hole?

 

[John232]

Length contraction is only supposed to work in the direction of motion so if it was 2m wide then no since it would be contracted in the wrong direction.

 

[jester1989]

the car's length is 2m, and the holes length is 1m, the hole's width>car's width.(since this doesn't contract. Car's direction of travel is parallel to the car's and hole's length.

 

[John232]

Sounds like you got length and width confused. If the direction of travel was into the hole then the side perpendicular to the direction of travel would be unaffected. This would be the side that would make the difference in being able to go in the hole or not, the unaffected side.

 

[bcrowell]

http://en.wikipedia.org/wiki/Ladder_paradox

 

[Dale]

the car's length is 2m, and the holes length is 1m, the hole's width>car's width.(since this doesn't contract. Car's direction of travel is parallel to the car's and hole's length.

The question is, in your thought experiment, under what conditions does the car fall into the hole. If you answer that question is a rigorous manner then you will find that the answer is the same in all frames.

 

[jester1989]

@john Sounds like you got length and width confused. If the direction of travel was into the hole then the side perpendicular to the direction of travel would be unaffected. This would be the side that would make the difference in being able to go in the hole or not, the unaffected side.

hmm..nt sure if i am getting this. so since in the question, the width is larger than the car(both width is unaffected)...and since the car length contracts, it will fit into the hole?


or...mb just change to 1 dimension..x... @@@@@@(car) -> ... (dots=small hole) will the car fit into the hole if it moves fast enough?

 

[jester1989]

or...paper thin car and hole..with negliable(or NO) width

 

[yuiop]

will a 2m(rest length) car moving near lightspeed fall in a 1m hole?

This is a fairly well known paradox, but it is not popular because it is very "messy" and has a number of practical issues that have to be resolved. Here are some of the issues outlined:

1) A car would normally have suspension, which would complicate the calculations, so I will take the liberty of replacing the car with a flat ice puck moving on a frictionless surface. The ice puck would have to be traveling at a velocity somewhat greater than 0.866c relative to the surface, to get the required length contraction. At that the velocity the time for the front edge of the ice puck to traverse the hole would be 1.926E-9 seconds. Assuming a downward gravitational force of 10m/s^2 the leading edge of the puck would fall a distance of 0.0000000000185 millimetres. That would hardly register as a bump in the pucks straight line path across the top of the hole. To make the puck fall a noticeable distance (say 1 cm) into the hole the gravity would have to be over 500,000,000,000 times greater than the Earth's surface gravity.

2) In the reference frame in which the puck is rest, the hole is only 0.5 meters wide and the car is 2 meters wide. The only way the puck can enter the hole is if the puck bends. This requires that the rigidity of the puck material has to be limited and as already mentioned, the gravitational force has to huge.

3) In the rest frame of the surface, there is point at which more than half of the puck is unsupported as it starts to cross the gap. At this point the edge of the hole acts a fulcrum and the puck starts to rotate. This rotation means the puck approaches the far side sideways on and is unlikely to cleanly pass under the far edge even if the puck and surface are paper thin. In the rest frame of the hole, there is never a point at which more than half of the puck is unsupported and there no reason why the puck should rotate in this frame. Presumably the rotation is replaced by the bending of the puck in this frame. This is one of the more messier aspects of the paradox.

4) It should be noted that the acceleration due to gravity is effectively greater (by gamma squared) in the frame in which the puck is at rest due to time dilation. For example if the puck falls 5m in one second in the rest frame of the surface then it falls 5m in less than than half a second in the puck frame.

So as you can see, in order to determine if the "vehicle" falls into the hole or not, we need to know the height of the vehicle and the thickness of the surface, the force of gravity, the tensile strength or rigidity of the material the vehicle is made of and the suspension arrangement (if any). All we can be sure of without knowing all those details is that whatever the fate of the vehicle in any given reference frame, that the same fate happens in any other reference frame. If it can be shown that different outcomes are predicted in different reference frames, then relativity is a broken theory or (more likely) we have not taken everything into account.

 

[WastedGunner]

This is a fairly well known paradox, but it is not popular because it is very "messy" and has a number of practical issues that have to be resolved. Here are some of the issues outlined:

1) A car would normally have suspension, which would complicate the calculations, so I will take the liberty of replacing the car with a flat ice puck moving on a frictionless surface. The ice puck would have to be traveling at a velocity somewhat greater than 0.866c relative to the surface, to get the required length contraction. At that the velocity the time for the front edge of the ice puck to traverse the hole would be 1.926E-9 seconds. Assuming a downward gravitational force of 10m/s^2 the leading edge of the puck would fall a distance of 0.0000000000185 millimetres. That would hardly register as a bump in the pucks straight line path across the top of the hole. To make the puck fall a noticeable distance (say 1 cm) into the hole the gravity would have to be over 500,000,000,000 times greater than the Earth's surface gravity.

2) In the reference frame in which the puck is rest, the hole is only 0.5 meters wide and the car is 2 meters wide. The only way the puck can enter the hole is if the puck bends. This requires that the rigidity of the puck material has to be limited and as already mentioned, the gravitational force has to huge.

3) In the rest frame of the surface, there is point at which more than half of the puck is unsupported as it starts to cross the gap. At this point the edge of the hole acts a fulcrum and the puck starts to rotate. This rotation means the puck approaches the far side sideways on and is unlikely to cleanly pass under the far edge even if the puck and surface are paper thin. In the rest frame of the hole, there is never a point at which more than half of the puck is unsupported and there no reason why the puck should rotate in this frame. Presumably the rotation is replaced by the bending of the puck in this frame. This is one of the more messier aspects of the paradox.

4) It should be noted that the acceleration due to gravity is effectively greater (by gamma squared) in the frame in which the puck is at rest due to time dilation. For example if the puck falls 5m in one second in the rest frame of the surface then it falls 5m in less than than half a second in the puck frame.

So as you can see, in order to determine if the "vehicle" falls into the hole or not, we need to know the height of the vehicle and the thickness of the surface, the force of gravity, the tensile strength or rigidity of the material the vehicle is made of and the suspension arrangement (if any). All we can be sure of without knowing all those details is that whatever the fate of the vehicle in any given reference frame, that the same fate happens in any other reference frame. If it can be shown that different outcomes are predicted in different reference frames, then relativity is a broken theory or (more likely) we have not taken everything into account.

The car is experiencing acceleration due to gravity, therefore it is not in an inertial frame and you cannot use special relativity.

Nothing is broken

 

[subhendu001]

i think the ans is NO...

bcuz d hole is also moving relative to the car.. so here also length contraction will take place...
am i clear to express my thought?

 

[harrylin]

will a 2m(rest length) car moving near lightspeed fall in a 1m hole?

From bcrowell's link, this paradox is discussed in detail in http://iopscience.iop.org/0143-0807/26/1/003.

And Yuiop summarized it rather well I think. Sticking with your car, the car's wheels (or, the surface of the tires) will go down a negligible amount before the car reaches the other end of the hole - ignoring the fact that the wheels would explode at such a speed.