| <!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="The scheduled departure time of the train is 12.00. It's 12.00 on your watch, but the penultimate car that moves past you during time t_1 is already starting to pass by you.The last car passes you during t_2. The train left on time and is moving equidistant. How far behind is your watch?"> | | <meta name="description" content="The scheduled departure time of the train is 12.00. It's 12.00 on your watch, but the penultimate car that moves past you during time t_1 is already starting to pass by you.The last car passes you during t_2. The train left on time and is moving equidistant. How far behind is your watch?"> |
| <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="The scheduled departure time of the train is 12.00. It's 12.00 on your watch, but the penultimate car that moves past you during time t_1 is already starting to pass by you.The last car passes you during t_2. The train left on time and is moving equidistant. How far behind is your watch?"> | | <meta property="og:title" content="The scheduled departure time of the train is 12.00. It's 12.00 on your watch, but the penultimate car that moves past you during time t_1 is already starting to pass by you.The last car passes you during t_2. The train left on time and is moving equidistant. How far behind is your watch?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="The scheduled departure time of the train is 12.00. It's 12.00 on your watch, but the penultimate car that moves past you during time t_1 is already starting to pass by you.The last car passes you during t_2. The train left on time and is moving equidistant. How far behind is your watch?"> | | <meta property="og:description" content="The scheduled departure time of the train is 12.00. It's 12.00 on your watch, but the penultimate car that moves past you during time t_1 is already starting to pass by you.The last car passes you during t_2. The train left on time and is moving equidistant. How far behind is your watch?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>The scheduled departure time of the train is 12.00. It's 12.00 on your watch, but the penultimate car that moves past you during time t_1 is already starting to pass by you.The last car passes you during t_2. The train left on time and is moving equidistant. How far behind is your watch?</title> | | <title>The scheduled departure time of the train is 12.00. It's 12.00 on your watch, but the penultimate car that moves past you during time t_1 is already starting to pass by you.The last car passes you during t_2. The train left on time and is moving equidistant. How far behind is your watch?</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="../../#1.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.2">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.2.21^*.$ The scheduled departure time of the train is $12.00$. It's $12.00$ on your watch, but the penultimate car that moves past you during time $t_1$ is already starting to pass by you. | | $1.2.21^*.$ The scheduled departure time of the train is $12.00$. It's $12.00$ on your watch, but the penultimate car that moves past you during time $t_1$ is already starting to pass by you. |
| The last car passes you during $t_2$. The train left on time and is moving equidistant. How far behind is your watch? | | The last car passes you during $t_2$. The train left on time and is moving equidistant. How far behind is your watch? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| $$l=\nu _{1}t_{1} +\frac{at_{1}^{2}}{2}$$ | | $$l=\nu _{1}t_{1} +\frac{at_{1}^{2}}{2}$$ |
| | | |
| $$l=(\nu _{1}+at_{1})t_{2} +\frac{at_{1}^{2}}{2}$$ | | $$l=(\nu _{1}+at_{1})t_{2} +\frac{at_{1}^{2}}{2}$$ |
| | | |
| $$(\nu _{1}+at_{1})t_{2} +\frac{at_{1}^{2}}{2} = \nu _{1}t_{1} +\frac{at_{1}^{2}}{2}$$ | | $$(\nu _{1}+at_{1})t_{2} +\frac{at_{1}^{2}}{2} = \nu _{1}t_{1} +\frac{at_{1}^{2}}{2}$$ |
| Given that $\nu _{1} = a{d}t$: | | Given that $\nu _{1} = a{d}t$: |
| $$a{d}t(t_{1}-t_{2}) =\frac{a(t_{2}-2t_{1}t_{2}-t_{1}^{2})}{2}$$ | | $$a{d}t(t_{1}-t_{2}) =\frac{a(t_{2}-2t_{1}t_{2}-t_{1}^{2})}{2}$$ |
| $${d}t=\fbox{$\frac{t_{2}-2t_{1}t_{2}-t_{1}^{2}}{2(t_{1}-t_{2})}$}$$ | | $${d}t=\fbox{$\frac{t_{2}-2t_{1}t_{2}-t_{1}^{2}}{2(t_{1}-t_{2})}$}$$ |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| <a href="https://www.instagram.com/_den.di__/" target="_blank">Dzikan Danila</a></b> | | <a href="https://www.instagram.com/_den.di__/" target="_blank">Dzikan Danila</a></b> |
| </p> | | </p> |
| | | |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\frac{t_{2}-2t_{1}t_{2}-t_{1}^{2}}{2(t_{1}-t_{2})}$$ | | $$\frac{t_{2}-2t_{1}t_{2}-t_{1}^{2}}{2(t_{1}-t_{2})}$$ |
| </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> |