Relativity: Calculating Separation r' between Emission and Reception

[Void123]

Homework Statement

A light pulse is emitted at a position $$x_{A}$$ (horizontally) and is received at position $$x_{B} = x_{A} + r$$. Considering that $$v = \beta c$$ for a moving reference frame, I must calculate the separation $$r'$$ between the point of emission and reception.

Homework Equations

$$x_{B} - x_{A} = \gamma(x_{B}' - x_{A}')$$

The Attempt at a Solution

I used the above equation, solving for $$x_{B}' - x_{A}'$$, but the answer provided by the book gives me an answer that differs from mine by a few operation signs. But I cannot think how they got it.

 

[vela]

Try writing down the spacetime coordinates in the rest frame for when the pulse is emitted and when it's received. Then use the Lorentz transformations to calculate the coordinates in the moving frame.

 

[Void123]

This is precisely what I have written above:

$$l_{moving} = \frac{l_{rest}}{\gamma}$$

 

[vela]

No, it isn't. That's the formula for length contraction, which isn't applicable for this problem because the emission and reception occur at different times.

 

[Void123]

Okay, for the time:

$$t_{2} = t_{1}'\sqrt{\frac{1 - \beta}{1 + \beta}}$$

Essentially, this is the doppler effect.

Can I rewrite the times in terms of $$l$$ and $$l'$$ to get the answer I want?