> College of Science > Physics Dept > Tatsu Takeuchi > Special Relativity > Practice Problems > Problem 14
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### Trains in a Tunnel

Two trains, 1 and 2, of equal lengths are traveling at the same speed in opposite directions inside a long, dark, and straight tunnel. At time t1 in the ground frame of reference, all the ceiling lights in the tunnel turn on simultaneously. Then, at time t2 in the ground frame of reference, they all turn off simultaneously.

In ground frame of reference, both trains are observed to be the same length (since they are both moving at the same speed) and both are immersed in light for the same amount of time. Therefore, an observer on the ground concludes that both trains are hit by the same number of photons.

However, an observer riding train 1 reasons that since train 1 is longer than train 2 in the train 1 frame of reference (since train 2 in shorter due to Lorentz contraction) train 1 must be hit by more photons than train 2. This seems to contradict the conclusion of the observer on the ground.

Answer the following questions:

1. At which spacetime points, labeled A through V, are the front and rear ends of train 1 when its front end enters the field of light in the train 1 frame of reference.
2. At which spacetime points, labeled A through V, are the front and rear ends of train 1 when its rear end exits the field of light in the train 1 frame of reference.
3. At which spacetime points, labeled A through V, are the front and rear ends of train 2 when its rear end enters the field of light in the train 1 frame of reference.
4. At which spacetime points, labeled A through V, are the front and rear ends of train 2 when its front end exits the field of light in the train 1 frame of reference.
5. What was wrong with the reasoning of the observer on train 1? Explain, referring to the spacetime diagram.
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