Redshift of a light pulse between 2 accelerating rockets

[sphyrch]

I'm reading book from here. Suppose two rockets are accelerating with the same acceleration $a$ and are separated by some distance $z$. At time $t_0$ the trailing rocket emits a light pulse. The book tells that pulse reaches leading box after time $z/c$ as seen in background frame. But won't the pulse actually have to cover a distance more than $z$ to reach the front rocket since the front rocket would've moved forward in that time? This on pg 65

 

[Dale]

Yes, that is a first order approximation.

 

[sphyrch]

Yes, that is a first order approximation.

It like this? If front ship moved extra $x$ dist by the time (say $t$) light reached, then $ct-z=ut+at^2/2$. and then we say $u<<c$ so we ignore, and we say that time taken is super short so we ignore $t^2$ too. So every thing gets ignored and we get $ct-z=0$. This the author logic?

 

[Dale]

More or less, yes. The only other thing is that usually they choose the reference frame where $u=0$. So the displacement due to acceleration is 2nd order ($at^2/2$)