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Old 2003-08-17, 00:37   #1
Wacky
 
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Default The Railroad Puzzles

Number 1:

Out in West Texas there are many miles of perfectly straight track.
Let us assume that some night we go out and select a section of perfectly welded track without any joints. We securely anchor each end of a section that is exactly 1 mile long. The next afternoon is a typical "scorcher" and the sun heats the track causing it to expand. Assume that the one mile of track is lengthened by only one foot (0.02%). Further assume that all the intermediate ties are loose and the track raises vertically to accomodate the extra length.

If we allowed the track the make abrupt bends, it could instantly jump up six inches and then abruptly drop six inches to include the extra length.

Or it could go up 3 inches, return to the normal elevation and make a second similar 3 inch step.

Or it could make 4 steps of 1.5 inches, etc.

But, steel rail does not have those characteristics. So assume that it raises up in a smooth circular arc. The high point will be at the middle and it will slowly decend toward each end.

The question is: "How high will the midpoint be above its nominal location?"
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Old 2003-08-17, 00:43   #2
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Number 2:

A freight train is exactly one mile long at rest.
If it is travelling at 60 miles per hour, how long will it be?

PS: This was an actual question on the final exam of a physics course.
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Old 2003-08-17, 01:06   #3
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I'll raise one question just for the fun of it. Years ago I was told that railroad track does not expand in the lengthwise direction rather it expands in width and height. Your posted question is based on the length expanding. Am I to presume that this is just a mental exercise, not a question based on fact?

On the second question, I presume you are wanting to apply Einsteins equations to the train.

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Old 2003-08-17, 01:27   #4
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Quote:
Originally Posted by Fusion_power
I'll raise one question just for the fun of it. Years ago I was told that railroad track does not expand in the lengthwise direction rather it expands in width and height. Your posted question is based on the length expanding. Am I to presume that this is just a mental exercise, not a question based on fact?
I think that your "fact" is in error. The steel does not "know" the distinction in "height", "width", and "length". It actually expands in all directions. But the 0.02% expansion in width and height amounts to thousandths of an inch. If is only in the "length" that the cumulative expansion is noticable.

Quote:
On the second question, I presume you are wanting to apply Einsteins equations to the train.
As I said, this was an actual question on the final exam for a physics class.

Yes, I think that you should consider "The Lorentz Transformation".
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Old 2003-08-17, 01:34   #5
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Quote:
Originally Posted by Fusion_power
I'll raise one question just for the fun of it. Years ago I was told that railroad track does not expand in the lengthwise direction rather it expands in width and height.
I think you were mis-informed. While it's conceivable that crystals might show differential expansion in different directions, there isn't any way for a cubic millimeter of steel in the middle of the rail to know which way it is oriented. Additionally, if you have ever walked along a railroad track you may have noticed the rails don't touch end to end - somebody thinks it's important to leave an expansion gap. Finally, if this were true, we would bridges out of railroad ties and save all the trouble of expansion joints.

Or perhaps the person was trying to say that the overall length of a multi-rail stretch does not change because the expansion gaps absorb the difference.
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Old 2003-08-17, 01:42   #6
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Quote:
Originally Posted by wblipp
Or perhaps the person was trying to say that the overall length of a multi-rail stretch does not change because the expansion gaps absorb the difference.
Note that I specified that it was uniformly welded steel rail. I assume that there are no expansion joints within the interval.

(And after you do the math, you too may wonder just how they make "welded rail" work. In olden days, the "clickity-clack of the railroad track" was because of those gaps.)
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Old 2003-08-18, 04:28   #7
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How could only going 60 MPH be enough to make the train any longer in a measureable sense? I thought Einstein's theories only dealt with speed at or near that of light.
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Old 2003-08-18, 04:59   #8
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In regards to puzzle 2:

Using a North American style frieght train, we could esimate how much longer the train would be.

When the train is stopped it is exactly 5280' long and all the couplers are in the "clopased" state.
As the train starts up the engine(s) begin to pull the couplers into the running position.
With a rough figure of 3" expansion per coupler.
For the purposes of the test figure the 'old fashion' 60' box cars (and engine).

This gives us 88 total cars (and engine). Giving 87 couplings.

87 x 3 = 261" = 21.75'

5280 + 21.75 = 5300.75' total length at any non relativistic speed.
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Old 2003-08-18, 05:46   #9
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Re the train question:

Presuming that the train is exactly 5280 feet long at rest and that its length is fixed i.e. it doesn't stretch due to couplings, etc.

If the train accelerates to 60 mph, it would shrink by about .0002 feet in length.

If it accelerated to 1/2 lightspeed, it would shrink much more dramatically

For all practical purposes, the train stays the same length as it approaches light speed, its frame of reference shrinks though so an outside observer thinks the train is shorter.

What happens when a relativistic object meets another relativistic object head on? What if a relativistic object (that weighed just one pound at rest) struck the earth doing .8C?

Relativistic object is defined as anything moving more than half the speed of light.
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Old 2003-08-18, 09:19   #10
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In terms of relativity the length of the train depends on the frame of reference used to measure it.
In the train's frame of reference it is always 1 mile long, no matter how it's moving.

In the frame of reference of the tracks, the train would be shorter by three thousands of an inch.

But the question doesn't specify which frame you're using so it's meaningless as an exam question (unless you want the answer "it depends").

Puzzles, of course, on the other hand, tend to over or under specify the conditions as part of the puzzle - in this case my answer is "I don't know".


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Old 2003-08-18, 10:25   #11
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Quote:
Originally Posted by Fusion_power
If the train accelerates to 60 mph, it would shrink by about .0002 feet in length.
I think you forgot to square v/c. My answer is that at 60 mph, it would shrink by about 2e-11 feet in length.
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