| <!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="An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity c = (c_x,c_y,c_z) falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the yOz plane? After reflection from all three mirrors?"> | | <meta name="description" content="An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity c = (c_x,c_y,c_z) falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the yOz plane? After reflection from all three mirrors?"> |
| <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="An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity c = (c_x,c_y,c_z) falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the yOz plane? After reflection from all three mirrors?"> | | <meta property="og:title" content="An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity c = (c_x,c_y,c_z) falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the yOz plane? After reflection from all three mirrors?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity c = (c_x,c_y,c_z) falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the yOz plane? After reflection from all three mirrors?"> | | <meta property="og:description" content="An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity c = (c_x,c_y,c_z) falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the yOz plane? After reflection from all three mirrors?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity c = (c_x,c_y,c_z) falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the yOz plane? After reflection from all three mirrors?</title> | | <title>An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity c = (c_x,c_y,c_z) falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the yOz plane? After reflection from all three mirrors?</title> |
| <link rel="stylesheet" href="../../css/css-latex/style.css"> | | <link rel="stylesheet" href="../../css/css-latex/style.css"> |
| <link rel="icon" href="../../img/logo.png" type="image/png"> | | <link rel="icon" href="../../img/logo.png" type="image/png"> |
| <script src="../../js/jquery-1.10.1.min.js"></script> | | <script src="../../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;"> |
| <div id = "logo"> | | <div id = "logo"> |
| <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> |
| $1.1.21.$ An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity $c = (c_x,c_y,c_z)$ falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the $yOz$ plane? After reflection from all three mirrors? | | $1.1.21.$ An angular reflector mounted on the moon rover consists of three mutually perpendicular mirrors. If light, whose velocity $c = (c_x,c_y,c_z)$ falls on the reflector. What components will have the velocity of light after reflection from the mirror located in the $yOz$ plane? After reflection from all three mirrors? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| For problem $1.1.21$ | | For problem $1.1.21$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| <p> | | <p> |
| $a)$ Since the mirror in the $yOz$ plane reflects the beam along the $Ox$ axis, the $x$ coordinate is inverted. | | $a)$ Since the mirror in the $yOz$ plane reflects the beam along the $Ox$ axis, the $x$ coordinate is inverted. |
| | | |
| </p> | | </p> |
| <p> | | <p> |
| Thus, if $\vec{c_1}$ is the reflected ray from $\vec{c}$, | | Thus, if $\vec{c_1}$ is the reflected ray from $\vec{c}$, |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $\vec{c_1} = (-c_x, c_y, c_z)$ | | $\vec{c_1} = (-c_x, c_y, c_z)$ |
| </p> | | </p> |
| <p> | | <p> |
| $b)$ Similarly, the mirrors in the $yOx$ and $xOz$ planes reflect the beam along the $Oz$ and $Oy$ axes, respectively: | | $b)$ Similarly, the mirrors in the $yOx$ and $xOz$ planes reflect the beam along the $Oz$ and $Oy$ axes, respectively: |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $\vec{c_2} = (c_x, c_y, -c_z)$, for$yOx$ | | $\vec{c_2} = (c_x, c_y, -c_z)$, for$yOx$ |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $\vec{c_3} = (c_x, -c_y, c_z)$, for $xOz$ | | $\vec{c_3} = (c_x, -c_y, c_z)$, for $xOz$ |
| </p> | | </p> |
| <p> | | <p> |
| At reflection from three mirrors at once, all components of $\vec{c}$ will take their negative values: | | At reflection from three mirrors at once, all components of $\vec{c}$ will take their negative values: |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $\vec{c_4} = (-c_x, -c_y, -c_z)$ | | $\vec{c_4} = (-c_x, -c_y, -c_z)$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$(-c_x,c_y,c_z), (-c_x,-c_y,-c_z)$$ | | $$(-c_x,c_y,c_z), (-c_x,-c_y,-c_z)$$ |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> |
| </p> | | </p> |
| <p> | | <p> |
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> | | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> |
| </p> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
