Special relativity, a train and a light pulse

AI Thread Summary
The discussion centers on a problem involving special relativity, where a photon emitted from a bulb must reflect off the roof of a room to reach a receptor while observed from a moving train. Two different calculations yield different time results for the photon to reach the receptor. The first approach incorrectly assumes the horizontal distance is contracted without considering the motion of the room relative to the train. The second approach, which uses Lorentz transformation, is deemed correct but requires careful consideration of both terms in the equation, as both can be significant. Ultimately, the first method is flawed due to its oversight of the room's motion in the train's frame.
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i am having some hard time thinking about this problem:

It is basically this:

Imagine a bulb and a receptor distant L from each other (at the same axis x) inside a room, the roof of the room is at a height d from the bulb and receptor. Now you are at a train moving horizontally, parallel to the x axis, with speed v. You are looking to the room. What time does it take in your reference frame to a photon emitted by the bulb to reach the receptor, in such way that the photon should reflect at the roof, and not be direct emitted along the closest distance between them.

The problem is that i am having two different answers, using different approach:

(1) The horizontal distance will be contract, so the time it take is $$t' = \frac{2\sqrt{(L/2\gamma)^2 + d^2}}{c}$$

(2) Using lorentz transformation, the time it take is $$t' = \gamma t - \gamma \beta \Delta x/c = \gamma (t -\Delta x \beta /c) = \gamma (2\sqrt{(L/2)^2 + d^2}/c - L \beta / c)$$

Now $$|\beta / c| << 1$$, so $$t' \approx \frac{2 \gamma \sqrt{(L/2)^2 + d^2}}{c} $$.

Since i should choose just one answer, i would say that the first one use more solid arguments, but still i can not say definitely if the first is in fact the right answer, or in another words, i am not able to refute the second answer.

So which one is right? what argument the wrong answer used? Thank you.
 
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You may be able to improve 1. The receptor is leaving so light has to travel more than the contracted train length in x.
 
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Can you make a diagram?
 
Herculi said:
So which one is right? what argument the wrong answer used?
The first method is wrong because it doesn't take into account the motion of the room in the train frame.

The second method is correct, but you cannot neglect the second term, as the first term may also be small.
 
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