Three microphones located on the same straight line at points A, B, C recorded successively at the moments t_A > t_B > t_C the sound of an explosion that occurred at point O, which lies on the segment AC. Find the length of the segment AO if AB = BC = L. At what point in time did the explosion occur? For problem 1.1.4

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Solutions of Savchenko Problems in Physics
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    <h3 id="back-link"><a href="/#1.1">$\leftarrow$Back</a></h3>

    <h3> Statement </h3>
    <p>
        $1.1.4.$ Three microphones located on the same straight line at points $A$, $B$, $C$ recorded successively at the moments $t_A > t_B > t_C$ the sound of an explosion that occurred at point $O$, which lies on the segment $AC$. Find the length of the segment $AO$ if $AB = BC = L$. At what point in time did the explosion occur?

    <h3>Solution</h3>
    <p>
                    <p>
          1. Let us introduce the following variables
        </p>
        <p>
         $$x_B=L-x$$
        </p>
        <p>
         $$t_A=t_B+\Delta t$$
        </p>
        <p>
          2. Time of flight of $\gamma$-quantum to counters
        </p>
        <p>
         $$t_B+\Delta t=\frac{x}{c}$$
        </p>
        <p>
         $$t_B=\frac{L-x}{c}$$
        </p>
        <p>
          3. Solving the equations together, we obtain
        </p>
        <p>
         $$\frac{L-x}{c} + \Delta t = \frac{x}{c}$$
        </p>
        <p>
         $$x= \frac{L+c\Delta t}{2}=1.15\\,м$$
        </p>
    </p>

    <h4>Answer</h4>
    <p>
        At a distance $1.15\\, \text{m}$ from the microphone $A$
    </p>


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