So you agree that in the unprimed frame, the primed clock at x'=k elapses $$\frac{-k}{v\gamma^2}$$ between t=0 and $$t=\frac{-k}{v\gamma}$$, yes?
Not a contradiction, chinglu, but relative measurements. Measurements in different frames don't necessarily agree with each other - you know that, right?
In this case, the relative difference means that you're comparing different intervals.
Both intervals end when x'=k coincides with x=0.
This event is not relative, because that's where the clock of interest is located - everyone agrees that the clock at x'=k reads t'=-k/v as it passes the x=0 marker.
But your starting event is relative:
You say the timer starts when the origins coincide. But this is relative, because the clock of interest is at a different location.
In the primed frame, time is defined by t'. The x and x' origins meet when t'=0, so the origins meet when t'=0 at x'=k.
But in the unprimed frame, time is defined by t, not t'. The x and x' origins meet when t=0, so the origins meet when t=0 at x'=k.
You can't start the timer twice, so you must choose one.
Which is it?
Does the timer start when t=0 at x'=k, or when t'=0 at x'=k?
