What is the wavelength of a beam sent between 2 spacecraft traveling towards each other at 0.5c?

Question

The speed of each spacecraft is 0.5c relative to a neutral observer, who observes each spacecraft traveling towards each other from opposite directions, at 0.5c each relative to the observer.

As measured from the sending spacecraft, the wavelength of the beam being send is 500nm.

What is the wavelength as measured at the receiving spacecraft.

Answer 1

The relative speed = (0.5c +0.5c)/ {1 + (0.5c*0.5c/c^2)}
= c/1.25 = 0.8c.

The length contracts by 500n X (1- 0.8*0.8) ^0.5
= 300nm.

Answer 2

Taking spacecraft A as the reference point, the observer is travelling towards the spacecraft at 0.5c and spacecraft B is travelling towards spacecraft A at 1c. Aside from the fact that this is impossible, the basic answer is that spacecraft A will never see spacecraft B or the light beam until the two spacecraft collide.

Answer 3

The second answerer correctly applied the velocity addition formula for special relativity to show that the relative velocity of the two spacecraft is 0.8c. However, his or her final answer was not correct. if ke is the wavelength of the source, the observed wavelength ko would be ke*sqrt((1 + v/c)/(1 - v/c)), where v is 0.8c in this case. So ko = (500 nm)*sqrt((1 + 0.8c/c)/(1 - 0.8c/c)) = (500 nm)*sqrt((1 + 0.8)/(1 - 0.8)) = (500 nm)*sqrt(1.8/0.2) = (500 nm)*sqrt(9) = (500 nm)*3 = 1500 nm.

Since the craft are moving apart, we know that the 500 nm wavelength will expand rather than contract. This is a quick inspection that shows us that 300 nm would not be the correct answer, while 1500 nm makes sense.