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# A supertrain (proper length 100 m) travels at a speed of 0.950c as it passes through a tunnel (proper length 50.0 m). As seen by a trackside observer, is the train ever completely within the tunnel? If so, how much space is there to spare?

## 18.775 $\mathrm{m}$

Gravitation

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##### Andy C.

University of Michigan - Ann Arbor

##### Marshall S.

University of Washington

##### Aspen F.

University of Sheffield

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### Video Transcript

this exercise. We have a train that has a proper length of 100 m and it's traveling with a speed of 0.95 times the speed of light relative to a tunnel. The 10 0 has a proper length of 50 m and we want to know if the train fits in the toe. According to an observer that ISS at rest with the ground, that means that it's at rest with the tunnel s well, notice that for this observer on the ground, the length of eternal is just the proper length, which is 50 m since the observers at rest with the total. Also remember that the link that an observer on the ground will measure for a moving observer is equal to the proper length. So let's let me write it a little differently, Uh, for the length and we measure quite observer on the ground, we'll be called D, and this is equal to the proper length of the movie object divided by gum where gamma is one over the square root of one minus. If he squared over C square. Okay, so the length of the train, as observed by the Observer on the ground is equal to, what 100 m. This is the proper length of the train times one minus 0.95 square and this is equal to 31.2 m. I noticed that this is smaller than the length of the tunnel, which is 50 m by a distance. So it's 15 minus. There. You won't wait too. So this is equal to 18.8 m. Okay, so not only does the train fit on the tunnel, but we have 18.8 m left.

#### Topics

Gravitation

##### Andy C.

University of Michigan - Ann Arbor

##### Marshall S.

University of Washington

##### Aspen F.

University of Sheffield

Lectures

Join Bootcamp