Which firecracker explodes first according to synchronized clocks on the ground?

[Eric_meyers]

Homework Statement

"Two firecrackers A and B are placed at x' = 0 and x' = 100 ns, respectively, on a train moving in the +x direction relative to the ground frame. According to synchronized clocks on the train, both firecrackers explode simultaneously. Which firecracker explodes first according to synchronized clocks on the ground? Explain.

Homework Equations

N/A

The Attempt at a Solution

Ok, so my logic is a person standing in the middle of the train will receive the light signal from A and B at the same time and hence will record the events as simultaneous. If you take that person and remove him from the train, A is moving towards him and B is moving away with him so thus it takes less "time" for light from A to hit that observer and thus A will be seen to explode first.

My question is, which would explode first would seem to depend on the position of the observer and from my understanding of relativity - the observation from the rest frame should be independent of the observer - is it something I'm missing from the definition of synchronized clocks?

 

[Doc Al]

Ok, so my logic is a person standing in the middle of the train will receive the light signal from A and B at the same time and hence will record the events as simultaneous.

OK.

If you take that person and remove him from the train, A is moving towards him and B is moving away with him so thus it takes less "time" for light from A to hit that observer and thus A will be seen to explode first.

You need to tighten up that reasoning. Imagine a second observer, on the platform. Let's say that those two observers pass each other at the exact moment that the light hits the observer on the train. Thus the light hits both observers at the same time.

Since A is moving towards that second observer, the light from A must travel a greater distance and take more time to reach him. Thus A must have exploded first. (According to the platform observers.)

My question is, which would explode first would seem to depend on the position of the observer and from my understanding of relativity - the observation from the rest frame should be independent of the observer - is it something I'm missing from the definition of synchronized clocks?

Why do you think the sequence of explosions depends on where the observer is? Note that what an observer literally 'sees' depends on where they are, but that's not what is meant by which exploded first. The observer uses his raw observations to deduce when the explosions must have occurred--all observers in the same frame will agree.

 

[Eric_meyers]

OK.

You need to tighten up that reasoning. Imagine a second observer, on the platform. Let's say that those two observers pass each other at the exact moment that the light hits the observer on the train. Thus the light hits both observers at the same time.

Since A is moving towards that second observer, the light from A must travel a greater distance and take more time to reach him. Thus A must have exploded first. (According to the platform observers.)
.

Wait, this I don't understand, if A is moving towards the observer by virtue of the train's movement, then wouldn't the distance between the observer and the point A be decreasing not increasing?

 

[Doc Al]

Wait, this I don't understand, if A is moving towards the observer by virtue of the train's movement, then wouldn't the distance between the observer and the point A be decreasing not increasing?

Yes, the distance between A and the observer is decreasing. But that means that when A exploded it was further away, so the light from that explosion must have traveled a greater distance to reach the observer than the light from B (which was even closer when it exploded).

 

[vela]

Wait, this I don't understand, if A is moving towards the observer by virtue of the train's movement, then wouldn't the distance between the observer and the point A be decreasing not increasing?

It gets kind of confusing, so I like to work backwards. So you have Joe on the train standing exactly midway between the two firecrackers. The light from each reaches him simultaneously, so he deduces the firecrackers must have exploded simultaneously.

Sally, who is on the ground, will agree that the light from both firecrackers reaches Joe simultaneously, but from her point of view, Joe is moving away from the light emitted by firecracker A but is moving toward the light emitted by firecracker B. If the light from both firecrackers is to reach Joe at the same time, firecracker A must have exploded first because it has to propagate over a longer distance before reaching him.