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| <meta name="author" content="Aliaksandr Melnichenka"> |
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| <title>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</title> |
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| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../#1.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../#1.1">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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? | | $1.1.4.$ 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? |
| | | |
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| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| For problem 1.1.4 | | For problem 1.1.4 |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p> | | <p> |
| 1. Let us introduce the following variables | | 1. Let us introduce the following variables |
| </p> | | </p> |
| <p> | | <p> |
| $$x_B=L-x$$ | | $$x_B=L-x$$ |
| </p> | | </p> |
| <p> | | <p> |
| $$t_A=t_B+\Delta t$$ | | $$t_A=t_B+\Delta t$$ |
| </p> | | </p> |
| <p> | | <p> |
| 2. Time of flight of $\gamma$-quantum to counters | | 2. Time of flight of $\gamma$-quantum to counters |
| </p> | | </p> |
| <p> | | <p> |
| $$t_B+\Delta t=\frac{x}{c}$$ | | $$t_B+\Delta t=\frac{x}{c}$$ |
| </p> | | </p> |
| <p> | | <p> |
| $$t_B=\frac{L-x}{c}$$ | | $$t_B=\frac{L-x}{c}$$ |
| </p> | | </p> |
| <p> | | <p> |
| 3. Solving the equations together, we obtain | | 3. Solving the equations together, we obtain |
| </p> | | </p> |
| <p> | | <p> |
| $$\frac{L-x}{c} + \Delta t = \frac{x}{c}$$ | | $$\frac{L-x}{c} + \Delta t = \frac{x}{c}$$ |
| </p> | | </p> |
| <p> | | <p> |
| $$x= \frac{L+c\Delta t}{2}=1.15\,м$$ | | $$x= \frac{L+c\Delta t}{2}=1.15\,м$$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| At a distance $1.15\, \text{m}$ from the microphone $A$ | | At a distance $1.15\, \text{m}$ from the microphone $A$ |
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