Measuring an Event — Question 8

Question 8

Three known light-path lengths determine a planar event position

Synchronized receivers at $(0,0)$, $(4,0)$ and $(0,4)$ metres register a flash with independently known emission time. Their inferred path-length squares are $r_0^2=5$, $r_x^2=5$, $r_y^2=13$ in $\mathrm m^2$.

1. Eliminate the quadratic terms to reconstruct the event position.

2. Check all three path lengths and explain why subtracting squared ranges is useful.

Question 8 - Solution

Three known light-path lengths determine a planar event position

(a) Method and calculation

$$r_x^2-r_0^2=(x-4)^2-x^2=16-8x,\quad\boxed{x=2\,\mathrm m}.$$$$r_y^2-r_0^2=(y-4)^2-y^2=16-8y,\quad\boxed{y=1\,\mathrm m}.$$

(b) Interpretation and further calculation

$$x^2+y^2=5,\quad(x-4)^2+y^2=5,\quad x^2+(y-4)^2=13.$$$$\boxed{(r_0,r_x,r_y)=(\sqrt5,\sqrt5,\sqrt{13})\,\mathrm m}.$$

Physical check

The differences linearize the coordinates; checking the original ranges tests consistency rather than merely solving the differences.