| | | |
| <!DOCTYPE html> | | <!DOCTYPE html> |
| <html lang="en"> | | <html lang="en"> |
| | | |
| <head> | | <head> |
| <meta charset="utf-8"> | | <meta charset="utf-8"> |
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | | <meta name="viewport" content="width=device-width, initial-scale=1.0"> |
| <meta http-equiv="content-language" content="en"> | | <meta http-equiv="content-language" content="en"> |
| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | | <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> |
| <meta name="description" content="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"> | | <meta name="description" content="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"> |
| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
| <meta property="og:title" content="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"> | | <meta property="og:title" content="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"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="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"> | | <meta property="og:description" content="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"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <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> | | <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> |
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | | <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> |
| <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> | | <link rel="icon" href="https://savchenkosolutions.com/img/logo.png" type="image/png"> |
| <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> | | <script src="https://savchenkosolutions.com/js/jquery-1.10.1.min.js"></script> |
| <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> | | <script async src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/MathJax.js?config=TeX-MML-AM_CHTML"></script> |
| <script type="text/x-mathjax-config"> | | <script type="text/x-mathjax-config"> |
| MathJax.Hub.Config({ | | MathJax.Hub.Config({ |
| extensions: ['tex2jax.js'], | | extensions: ['tex2jax.js'], |
| jax: ['input/TeX', 'output/HTML-CSS'], | | jax: ['input/TeX', 'output/HTML-CSS'], |
| tex2jax: { | | tex2jax: { |
| inlineMath: [['$', '$'], ['$', '$']], | | inlineMath: [['$', '$'], ['$', '$']], |
| processEscapes: true, | | processEscapes: true, |
| processClass: 'tex2jax', | | processClass: 'tex2jax', |
| ignoreClass: 'html' | | ignoreClass: 'html' |
| }, | | }, |
| showProcessingMessages: false, | | showProcessingMessages: false, |
| messageStyle: 'none' | | messageStyle: 'none' |
| }); | | }); |
| </script> | | </script> |
| </head> | | </head> |
| <body style=""> | | <body style=""> |
| <header style="text-align:center;"> | | <header style="text-align:center;"> |
| <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> |
| | | |
| <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? |
| | | |
| <center> | | <center> |
| <figure> | | <figure> |
| <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 $A$ |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
