Who wins the relativistic space race?

[InsertName]

Homework Statement

OK, so spaceship A and spaceship B are at the same location. Spaceship A sets off (instantaneously) at a speed v_A$$\approx$$c, traveling in a straight line towards the finish line, which will take him a time t_A.

After a time $$\frac{t_A}{2}$$ (in spaceship B's frame of reference), spaceship B sets off towards the finish line at a speed v_B$$\approx$$c, which will get him there in a time $$\frac{t_A}{2}$$ (in his frame of reference).

Who, if anybody, wins the race?

Homework Equations

I'm not sure if equations are actually needed here as this is kind of a thought experiment.

The Attempt at a Solution

I have thought about this for a long time and can't come to a conclusion.

 

[ideasrule]

Unless this is a trick question, the solution seems very simple. Let's imagine an observer at the start position who never moves. He sees A head off at v=c, and B head off at the same speed a while later. A obviously wins the race. The result is the same in all reference frames because all reference frames must agree on the events that occur--it wouldn't make physical sense if one person thinks A crashed through the finish line while somebody else thinks it was B.

The problem becomes more interesting when you consider what happens in the spacecraft s' reference frames. B must be going faster because it arrives in only ta/2, meaning it sees the distance between start and finish as half what A believes it to be. Both A and B think that A starts off before B, that B is catching up, but that it will not have time to catch up before the finish line. If you have time, you can try to justify this rigorously using Lorentz transforms and the velocity addition formula.

 

[InsertName]

Thanks.