How Does Relative Motion Affect Light Travel Time in Special Relativity?

[Sabrewolf]

1. A spaceship is moving directly toward a planet at a speed of c/2. When the spaceship is 4.5E8 m from the planet a pulse of light is emitted by someone on the planet. As measured by someone on the spaceship, how long does it take the light pulse to travel from the planet to the ship?



2. I'm sure there are equations, but this question (from the explanation) relies upon Einstein's 2nd postulate of special relativity: That the speed of light is a universal constant regardless of the motion of the source or observer. The book states that because of this the occupants on the ship will still measure the speed of light to be c even though they are moving. As such the time it takes the pulse to travel 4.5E8 will be 1.5s given c=3E8 m/s



3. I accept that the speed of light is constant at 3E8 m/s. However my problem lies with the fact that the ship is moving TOWARDS the planet at half that speed. After 1 second, the pulse of light would have traveled 3E8 m as the beam will travel at the speed c, however the ship would have also moved towards the planet 1.5E8 m as it is also traveling towards the planet. After 1 second, wouldn't the ship and the pulse of light meet each other? I'm not sure if the book isn't taking into account the distance the ship moves or if I'm just not getting it.

 

[zhermes]

Yeah, no, you're absolutely right. Maybe the text is just trying to make it as simple as possible? (Also, its important to know which reference frame measures the 4.5e8 meters. From the answer, 4.5e8m is measures from the ship itself----but that's not at all obvious in the question).

 

[Sabrewolf]

Also, its important to know which reference frame measures the 4.5e8 meters

The book says that this is measured from an occupant on the ship. Would that affect the answer at all?

 

[Janus]

The book says that this is measured from an occupant on the ship. Would that affect the answer at all?

Yes. As far as the occupant is concerned, he can treat this problem as if he was stationary and the planet was moving towards him.

 

[Sabrewolf]

Yes. As far as the occupant is concerned, he can treat this problem as if he was stationary and the planet was moving towards him.

So does this change the actual answer? As far as I can see the answer is either 1.5s or 1s at this point, but I don't think the book takes into account the distance the ship travels in the 1 second that passes.

 

[Janus]

So does this change the actual answer? As far as I can see the answer is either 1.5s or 1s at this point, but I don't think the book takes into account the distance the ship travels in the 1 second that passes.

You are missing the meaning behind the second postulate. If the occupant of the ship measures the speed of light to be 3e8 m/s relative to himself, and the distance to the point from which the light was emitted from is 4.5e8 m as measured by him, then the light takes 1.5 sec to reach the ship. There is "no distance that the ship travels in one second" because the ship is at rest with respect to the occupant.

The book is correct and takes everything into account that it needs to.

 

[Sabrewolf]

You are missing the meaning behind the second postulate. If the occupant of the ship measures the speed of light to be 3e8 m/s relative to himself, and the distance to the point from which the light was emitted from is 4.5e8 m as measured by him, then the light takes 1.5 sec to reach the ship. There is "no distance that the ship travels in one second" because the ship is at rest with respect to the occupant.

The book is correct and takes everything into account that it needs to.

I apologize, I am totally confused regarding this. Is there any way you could dumb it down even further? The way I'm interpreting the problem, the only way that the time could be 1.5 seconds was if the ship was at rest relative to the planet. I'm close to getting it though. I believe I'm on the fringe of understanding what you're saying, but I can't get over the edge.

 

[Janus]

I apologize, I am totally confused regarding this. Is there any way you could dumb it down even further? The way I'm interpreting the problem, the only way that the time could be 1.5 seconds was if the ship was at rest relative to the planet.

Why? From the perspective of the Ship, what happens to the distance between planet and ship after the light is emitted makes no difference. Imagine that there is a second ship at rest with respect to the first. It is 4.5e8 m away from the first ship in the direction of the planet, so that it is even with the planet when the light emitted.
Now imagine that it emits its own light at the same instant. So should the light from the second ship reach the first ship in 1.5 sec since the two ships are at rest with respect to each other, while the light from the planet takes 1 sec, even though both lights were emitted at the same time and from the same point? wouldn't that be a violation of the second postulate?

I'm close to getting it though. I believe I'm on the fringe of understanding what you're saying, but I can't get over the edge.

From the ship's perspective:

1.The ship and planet are 4.5e8 m apart.
2. The light is emitted by the planet.
3. The ship measures the light as moving at 3e8 m/s relative to the ship.
4. The ship measures the planet as following the light at 1.5e8 m/s

The only thing that effects the time that the light takes to reach the ship as seen by the ship is the distance from the ship at which the light was emitted, and the speed of light.

 

[Sabrewolf]

The only thing that effects the time that the light takes to reach the ship as seen by the ship is the distance from the ship at which the light was emitted, and the speed of light.

Okay so any movement after the light is transmitted is no factor? Even if the ship is approaching the beam of light directly or moving away?