Added 2.2.37

astrosander правка от
Посмотреть решение История
правка #2508 GitHub 6c037b4 позже →
en/2.2.37.md +136 −0
@@ -0,0 +1,136 @@
+<!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="solutions, savchenko, physics problems, olympiad physics, physics book">
+ <meta name="description" content="A website with solutions to physics problems from Savchenko Textbook">
+ <meta property="og:title" content="Savchenko Solutions">
+ <meta property="og:image" content="img/logo.png">
+ <meta property="og:description" content="A website with solutions to physics problems from Savchenko Textbook">
+ <meta name="yandex-verification" content="6cfda41f74038368">
+ <title>Savchenko Solutions</title>
+ <link rel="stylesheet" href="https://savchenko-physics.github.io/css/css-latex/style.css">
+ <link rel="icon" href="https://savchenko-physics.github.io/img/logo.png" type="image/png">
+ <script src="https://savchenko-physics.github.io/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 Physics Textbook</h2>
+ <p class="author">
+ Aliaksandr Melnichenka <br/>
+ October 2023
+ </p>
+ </header>
+
+ <h3 id="back-link"><a href="../#2.2">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $2.2.37.$ Determine the thrust force of an air-jet engine of an airplane flying at a speed of $V$. The mass consumption of fuel and air entering the engine is equal to $\mu_1$ and $\mu_2$, respectively. Velocity of combustion products relative to the aircraft at the engine exit $u$.
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+
+</p>
+<center>
+ <figure>
+ <img src="2.2.37.jpg"
+ loading="lazy" width="250" />
+ <figcaption>
+ Reference frame relative to the earth
+ </figcaption>
+ </figure>
+</center>
+<p>
+
+\begin{equation*}
+\Delta p_x = ((\mu_1 + \mu_2)u - \mu_2 v) \Delta t
+\end{equation*}
+
+From the condition:
+$$
+u_{\text{rel}} = u, \quad v_{\text{rel}} = 0
+$$
+
+Write the vector sum for the relative velocities
+$$
+\vec{v}_{\text{rel}} = \vec{0} - \vec{v}
+$$
+
+$$
+v_{\text{rel} \, x} = -v_x
+$$
+
+$$
++v_{\text{rel}} = -(-v) = v
+$$
+
+Find the change in momentum over a soapy time interval
+
+\begin{equation*}
+\Delta p_x = \left( \mu_1 u + \mu_2 (u - v) \right) \Delta t
+\end{equation*}
+
+Let's use Newton's second law in impulse form
+
+\begin{equation*}
+F_x = \frac{\Delta p_x}{\Delta t} = \mu_1 u + \mu_2 (u - v)
+\end{equation*}
+
+By definition of mass consumption
+
+$$
+m_1 = \mu_1 \Delta t, \quad m_2 = \mu_2 \Delta t
+$$
+
+According to Newton's third law, these forces are equal to
+
+\begin{equation*}
+T = -F_x
+\end{equation*}
+From where the required thrust force
+$$
+\boxed{|T| = \mu_1 u + \mu_2 (u - v)}
+$$
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $$F = \mu_2 (u - v) + \mu_1 u$$
+ </p>
+ <p style="text-align: right; font-style: italic; font-size: 14;">
+ Almaskhan Arsen<br>
+ </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>