Help Applying SR to Calculate Time Dilation

[Palpatine]

I made this pic to illustrate my problem.

A spaceship (labeled B) departs from a space station (labeled A) at 80% the speed of light (0.8c). A applies the time dilation principle and calculates that B's clocks are running slower than his.

Some time later a smaller craft (labeled C) detaches from B and travels at 80% the speed of light in the opposite direction. B applies the time dilation principle and calculates that C's clocks are running slower than his.

But this results in an illogical situation. B's clocks are slower than A's and C's clocks are slower than B's but C is stationary in A's reference frame so C's clocks are running at the same rate as A's.

What am i doing wrong?

 

[Orodruin]

You are implicitly assuming the existence of an absolute simultaneity. This no longer holds in relativity. Instead, you must be aware that events that are simultaneous for one observer are not necessarily simultaneous for another. Since the rate at which clocks tick in an inertial frame is based on the notion of simultaneity in that frame, you will get different results as different frames have different definitions of simultaneity.