Translated 1.3.3-1.3.15; Fixed GUI image generator
en/1.3.9.md
+92 −0
| @@ -0,0 +1,92 @@ | |||
| + | <!DOCTYPE html> | ||
| + | <html lang="en"> | ||
| + | |||
| + | <head> | ||
| + | <meta charset="utf-8"> | ||
| + | <meta name="viewport" content="width=device-width, initial-scale=1.0"> | ||
| + | <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="description" content="At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration a? The distance from the gun to the rocket launch site is L, the gun fires at an angle of 45^\circ to the horizon."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration a? The distance from the gun to the rocket launch site is L, the gun fires at an angle of 45^\circ to the horizon."> | ||
| + | <meta property="og:image" content="img/logo.png"> | ||
| + | <meta property="og:description" content="At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration a? The distance from the gun to the rocket launch site is L, the gun fires at an angle of 45^\circ to the horizon."> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration a? The distance from the gun to the rocket launch site is L, the gun fires at an angle of 45^\circ to the horizon.</title> | ||
| + | <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"> | ||
| + | <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 type="text/x-mathjax-config"> | ||
| + | MathJax.Hub.Config({ | ||
| + | extensions: ['tex2jax.js'], | ||
| + | jax: ['input/TeX', 'output/HTML-CSS'], | ||
| + | tex2jax: { | ||
| + | inlineMath: [['$', '$'], ['$', '$']], | ||
| + | processEscapes: true, | ||
| + | processClass: 'tex2jax', | ||
| + | ignoreClass: 'html' | ||
| + | }, | ||
| + | showProcessingMessages: false, | ||
| + | messageStyle: 'none' | ||
| + | }); | ||
| + | </script> | ||
| + | </head> | ||
| + | <body style=""> | ||
| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.3.9.$ At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration $a$? The distance from the gun to the rocket launch site is $L$, the gun fires at an angle of $45^\circ$ to the horizon. | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | Let's use the formula for the $x$ and $y$ coordinates obtained in <a href="../1.3.6" target="_blank">1.3.6</a>: | ||
| + | $$ y(t) = \frac{1}{\sqrt{2}}vt - \frac{gt^2}{2} $$ | ||
| + | |||
| + | $$ x(t) = \frac{1}{\sqrt{2}}vt $$ | ||
| + | In this case, the equation describing the rocket’s motion is: | ||
| + | $$ y(t) = \frac{at^2}{2} $$ | ||
| + | We write down the meeting conditions: | ||
| + | $$ \frac{1}{\sqrt{2}}vt - \frac{gt^2}{2} = \frac{at^2}{2}\;(1) $$ | ||
| + | |||
| + | $$ \frac{1}{\sqrt{2}}vt = L $$ | ||
| + | Where, moment of meeting: | ||
| + | $$ t = \frac{\sqrt{2}L}{v} $$ | ||
| + | |||
| + | $$ L=\frac{v\sqrt{2}}{g} $$ | ||
| + | Substitute into $(1)$: | ||
| + | $$ \frac{v\sqrt{2}}{g} = \frac{(a+g)v^2}{g} $$ | ||
| + | We obtain the required speed: | ||
| + | $$ \fbox{$ v=\sqrt{L(a+g)} $} $$ | ||
| + | |||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$v=\sqrt{L(a+g)}$$ | ||
| + | </p> | ||
| + | |||
| + | |||
| + | <footer class="row container"> | ||
| + | <br> | ||
| + | <p> | ||
| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
| + | </p> | ||
| + | <p> | ||
| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small> | ||
| + | </p> | ||
| + | </footer> | ||
| + | </body> | ||
| + | |||
| + | </html> | ||
| @@ -0,0 +1,92 @@ | |||
| <!DOCTYPE html> | |||
| <html lang="en"> | |||
| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
| <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="description" content="At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration a? The distance from the gun to the rocket launch site is L, the gun fires at an angle of 45^\circ to the horizon."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration a? The distance from the gun to the rocket launch site is L, the gun fires at an angle of 45^\circ to the horizon."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration a? The distance from the gun to the rocket launch site is L, the gun fires at an angle of 45^\circ to the horizon."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration a? The distance from the gun to the rocket launch site is L, the gun fires at an angle of 45^\circ to the horizon.</title> | |||
| <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"> | |||
| <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 type="text/x-mathjax-config"> | |||
| MathJax.Hub.Config({ | |||
| extensions: ['tex2jax.js'], | |||
| jax: ['input/TeX', 'output/HTML-CSS'], | |||
| tex2jax: { | |||
| inlineMath: [['$', '$'], ['$', '$']], | |||
| processEscapes: true, | |||
| processClass: 'tex2jax', | |||
| ignoreClass: 'html' | |||
| }, | |||
| showProcessingMessages: false, | |||
| messageStyle: 'none' | |||
| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.3.9.$ At what speed should a projectile fly out of a cannon at the moment of rocket launch in order to hit a rocket starting vertically with acceleration $a$? The distance from the gun to the rocket launch site is $L$, the gun fires at an angle of $45^\circ$ to the horizon. | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| Let's use the formula for the $x$ and $y$ coordinates obtained in <a href="../1.3.6" target="_blank">1.3.6</a>: | |||
| $$ y(t) = \frac{1}{\sqrt{2}}vt - \frac{gt^2}{2} $$ | |||
| $$ x(t) = \frac{1}{\sqrt{2}}vt $$ | |||
| In this case, the equation describing the rocket’s motion is: | |||
| $$ y(t) = \frac{at^2}{2} $$ | |||
| We write down the meeting conditions: | |||
| $$ \frac{1}{\sqrt{2}}vt - \frac{gt^2}{2} = \frac{at^2}{2}\;(1) $$ | |||
| $$ \frac{1}{\sqrt{2}}vt = L $$ | |||
| Where, moment of meeting: | |||
| $$ t = \frac{\sqrt{2}L}{v} $$ | |||
| $$ L=\frac{v\sqrt{2}}{g} $$ | |||
| Substitute into $(1)$: | |||
| $$ \frac{v\sqrt{2}}{g} = \frac{(a+g)v^2}{g} $$ | |||
| We obtain the required speed: | |||
| $$ \fbox{$ v=\sqrt{L(a+g)} $} $$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$v=\sqrt{L(a+g)}$$ | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||