Translated 1.3.3-1.3.15; Fixed GUI image generator

astrosander правка от 9 сент. 2024 г., 01:06

правка #4103 GitHub 3ea294d позже →

en/1.3.8.md +105 −0

		@@ -0,0 +1,105 @@
		+	<!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="A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is v, the angle of inclination of the mountain is \alpha, and the angle of fire relative to the horizon is \beta?">
		+	<meta name="author" content="Aliaksandr Melnichenka">
		+	<meta name="date" content="2023-10" scheme="YYYY-MM">
		+	<meta property="og:title" content="A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is v, the angle of inclination of the mountain is \alpha, and the angle of fire relative to the horizon is \beta?">
		+	<meta property="og:image" content="img/logo.png">
		+	<meta property="og:description" content="A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is v, the angle of inclination of the mountain is \alpha, and the angle of fire relative to the horizon is \beta?">
		+	<meta name="yandex-verification" content="6cfda41f74038368">
		+	<title>A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is v, the angle of inclination of the mountain is \alpha, and the angle of fire relative to the horizon is \beta?</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>
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		+	<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>
		+	$1.3.8.$ A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is $v$, the angle of inclination of the mountain is $\alpha$, and the angle of fire relative to the horizon is $\beta$?
		+	</p>
		+	<center>
		+	<figure>
		+	<img src="statement.png"
		+	loading="lazy" width="200" />
		+	<figcaption>
		+	For problem $1.3.8$
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+	</p>
		+
		+	<h3>Solution</h3>
		+	<p>
		+	The equations of motion of the projectile can be written as follows:
		+	$$ {x}={v}_{0}{t}\cos{\beta} $$
		+
		+	$$ {y}={v}_{0}{t}\sin{\beta}-\frac{{g}{t}^{2}}{2} $$
		+	Substitute into the equations the coordinates of the target $x = L; \;y = L \tan \alpha$
		+	$$ {L=v_{0}t\cos\beta} $$
		+
		+	$$ Ltg\alpha=v_{0}t\sin\beta-\frac{gt^{2}}{2} $$
		+	Let us express time from the first equation of the last system of equations and substitute its value into the second equation
		+	$$ {t=\frac{L}{v_{0}\cos\beta}} $$
		+
		+	$$ {Ltg\alpha=v_{0}\frac{L}{v_{0}\cos\beta}sin\beta-\frac{g}{2}\frac{L^{2}}{v_{0}^{2}\cos^{2}\beta} } $$
		+	Where
		+	$$ {v}_{0}=\sqrt{\frac{{gL}\cos\alpha}{2\cos\beta\sin(\beta-\alpha)}} $$
		+	We express $L$,
		+	$$ L = \frac{ 2\cos\beta\sin(\beta-\alpha)\cdot v^2_0}{g\cos\alpha} $$
		+	And we find the flight range along the wall:
		+	$$ l = \frac{L}{\cos \alpha} $$
		+
		+	$$ \fbox{$l = \frac{ 2v^2_0}{g} \frac{ \cos\beta\sin(\beta-\alpha)}{\cos^2\alpha}$} $$
		+
		+	</p>
		+
		+	<h4>Answer</h4>
		+	<p>
		+	$$L=\frac{2v^2}g\frac{\cos^2\beta}{\cos\alpha}(\text{tg}\beta-\text{tg}\alpha)$$
		+	</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 +1,105 @@
			<!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="A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is v, the angle of inclination of the mountain is \alpha, and the angle of fire relative to the horizon is \beta?">
			<meta name="author" content="Aliaksandr Melnichenka">
			<meta name="date" content="2023-10" scheme="YYYY-MM">
			<meta property="og:title" content="A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is v, the angle of inclination of the mountain is \alpha, and the angle of fire relative to the horizon is \beta?">
			<meta property="og:image" content="img/logo.png">
			<meta property="og:description" content="A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is v, the angle of inclination of the mountain is \alpha, and the angle of fire relative to the horizon is \beta?">
			<meta name="yandex-verification" content="6cfda41f74038368">
			<title>A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is v, the angle of inclination of the mountain is \alpha, and the angle of fire relative to the horizon is \beta?</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>
			$1.3.8.$ A mortar is fired at objects located on the mountainside. At what distance from the mortar will the mines fall if their initial velocity is $v$, the angle of inclination of the mountain is $\alpha$, and the angle of fire relative to the horizon is $\beta$?
			</p>
			<center>
			<figure>
			<img src="statement.png"
			loading="lazy" width="200" />
			<figcaption>
			For problem $1.3.8$
			</figcaption>
			</figure>
			</center>
			<p>
			</p>

			<h3>Solution</h3>
			<p>
			The equations of motion of the projectile can be written as follows:
			$$ {x}={v}_{0}{t}\cos{\beta} $$

			$$ {y}={v}_{0}{t}\sin{\beta}-\frac{{g}{t}^{2}}{2} $$
			Substitute into the equations the coordinates of the target $x = L; \;y = L \tan \alpha$
			$$ {L=v_{0}t\cos\beta} $$

			$$ Ltg\alpha=v_{0}t\sin\beta-\frac{gt^{2}}{2} $$
			Let us express time from the first equation of the last system of equations and substitute its value into the second equation
			$$ {t=\frac{L}{v_{0}\cos\beta}} $$

			$$ {Ltg\alpha=v_{0}\frac{L}{v_{0}\cos\beta}sin\beta-\frac{g}{2}\frac{L^{2}}{v_{0}^{2}\cos^{2}\beta} } $$
			Where
			$$ {v}_{0}=\sqrt{\frac{{gL}\cos\alpha}{2\cos\beta\sin(\beta-\alpha)}} $$
			We express $L$,
			$$ L = \frac{ 2\cos\beta\sin(\beta-\alpha)\cdot v^2_0}{g\cos\alpha} $$
			And we find the flight range along the wall:
			$$ l = \frac{L}{\cos \alpha} $$

			$$ \fbox{$l = \frac{ 2v^2_0}{g} \frac{ \cos\beta\sin(\beta-\alpha)}{\cos^2\alpha}$} $$

			</p>

			<h4>Answer</h4>
			<p>
			$$L=\frac{2v^2}g\frac{\cos^2\beta}{\cos\alpha}(\text{tg}\beta-\text{tg}\alpha)$$
			</p>


			<footer class="row container">
			<br>
			<p>
			<small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
			</p>
			<p>
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