Spaceship flying towards earth at relativistic speeds?

[novop]

Here is something I thought about the other day that could use some clarification... say a spaceship starts its voyage towards the Earth at 0.5c at a distance of 1 light year away from the earth. All measurements are made from an observer on earth.

An observer on Earth only notices the ship after 1 year has passed, but at this time the spaceship is in reality halfway between it's starting position and the earth. After 1.5 years have passed, the ship appears to be at the halfway position but is in reality 3/4 between it's starting position and the earth. To an observer on earth, wouldn't the ship have appeared to be approaching at the speed of light? Only half a year has passed but the image of the ship appears to have traversed a distance of half a light year. Where am I going wrong here?

 

[Ich]

Where am I going wrong here?

Nowhere. This effect is well-known, and has been observed for example in quasar jets. We had a https://www.physicsforums.com/showthread.php?p=2696616" about it recently.

 

[novop]

I see... so it is only infinitely close to the observer that the real position of the ship and it's trailing image converge to the same point?

 

[Ich]

Yes.

 

[novop]

Perfect thank you!