What time will it be before starship hits the mirror?

Question

This is silly that I have to repost this question, because Y!A has deleted my previous one. If at noon, a starship travelling at speed (1/2)c, sees a reflection of its own clock in a stationary mirror showing 11:00 am, what time will it be when the starship strikes the mirror?

Answer 1

12:30. I gave the explanation once. I'll do it again when I get a sec.

Okay, I'll work in my frame. My velocity is v. I read that my reflection time lags my watch by a time T.

During the time T, I have traveled a distance vT. The light forming the image has traveled a distance vT + 2d, where d is my distance from the mirror. The time the light took to do this was (vT+2d)/c, which of course is T.

So d = T(c-v)/2

I will cover that distance in a time t = d/v

t = T (c/v - 1)/2

If T = 1 hour and v = c/2, then t = 30 minutes.

Naively, it seems like the image's time is running 3 times fast, when conventional time dilation says it runs slow at 3/5 speed since the relative velocity is 4/5 c. But a velocity of 4/5c induces a 5x Doppler blueshift, so we get the watch seeming to run 3x as fast.

Answer 2

I preferred the previous question :(
At the same time you'd run into yourself!