Understanding the formula in an exercise

I'm stuck for quite some time now on this issue.

I have the following question.

Spaceship with velocity of $0.5c$ passes above two points $A$ and $B$ the distance between $A$ and $B$ respect to the ground is $80meter$.

A viewer on the ground measures the difference between $t_b$ and $t_a$ $t_a$ is the time that the spaceship passes above point $A$ and $t_b$ is the time that the spaceship passes above point $B$.

Somehow in the answers of this exercise they managed to reach to following formula : $$t_b-t_a=\frac{x_b-x_a}{v}=\frac{80}{0.5c}$$ which is not clear to me.

This is what I did:

According to lorentz transformation we know that

$$x_b=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}(x'+vt_b') \rightarrow 80=\frac{1}{\sqrt{1-\frac{0.25c^2}{c^2}}}(0+0.5ct_b')\rightarrow \frac{\sqrt{0.75}*80}{0.5c}$$

But that's not the formula/answer $t_b-t_a=\frac{x_b-x_a}{v}=\frac{80}{0.5c}$ Any ideas what's wrong or how to get to the formula/answer $t_b-t_a=\frac{x_b-x_a}{v}=\frac{80}{0.5c}$

Any help will be appreciated.

• This isn't an answer but a hint: what frame or frames are the measurements being done in? When do you need to use the Lorentz transformation? – tfb Mar 25 '16 at 21:43
• @tfb I'm not completely sure if this is what you mean but the measurements are in MKS unit system, and I use Lorentz transformation to calculate the change (time,location) between two inertial systems with different point of view EDIT : when the speed is very high to the speed of light ($c$) – John C Mar 25 '16 at 21:48
• @tfb OHHH, I think I understand what you mean, you mean that in that case I don't need to use Lorentz transformation but instead use the "normal" (not sure about the official name) which is $x=vt$ my question is why in this case I don't need to use Lorentz transformation? – John C Mar 25 '16 at 21:59
• @JohnC: You are most of the way there. The thing to understand is why you don't need to use the transformation, which is why I asked the question about which frames of reference (or frame of reference, hint) matter here. – tfb Mar 25 '16 at 22:27