1.1.4. Counters $A$ and $B$, which register the moment of the arrival of a $\gamma$-quantum, are located at a distance of $2\,\mathrm{m}$ from each other. At a point between them a $\pi_0$ meson decayed into two $\gamma$-quanta. Find the position of this point if counter $A$ detected the $\gamma$-quantum $10^{-9}\,\mathrm{s}$ later than counter $B$. The speed of light is $3\times10^8\,\mathrm{m/s}$.
Solution
1. Let us introduce the following variables
$$x_B=L-x$$
$$t_A=t_B+\Delta t$$
2. Time of flight of $\gamma$-quantum to counters
$$t_B+\Delta t=\frac{x}{c}$$
$$t_B=\frac{L-x}{c}$$
3. Solving the equations together, we obtain
$$\frac{L-x}{c} + \Delta t = \frac{x}{c}$$
$$x= \frac{L+c\Delta t}{2}=1.15\text{ m}$$
Answer
At a distance $1.15\text{ m}$ from the microphone $A$
