| <!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="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?"> | | <meta name="description" content="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?"> |
| <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="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?"> | | <meta property="og:title" content="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?"> | | <meta property="og:description" content="It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?</title> | | <title>It is raining heavily. Drop rate u. The ball slides on the asphalt at a speed of v. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round?</title> |
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | | <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"> | | <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 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 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;"> |
| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.4.17.$ It is raining heavily. Drop rate $u$. The ball slides on the asphalt at a speed of $v$. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round? | | $1.4.17.$ It is raining heavily. Drop rate $u$. The ball slides on the asphalt at a speed of $v$. How many times in the same period of time does it get more drops than the same, but stationary ball? Does the answer change if the ball is not round? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>During time $t$, all the drops that are in a cylinder of height $ut$ and base area $S_1$ equal to the area of the central cross-section of the ball perpendicular to $u$ fall on a stationary ball. That is, in the first case</p> | | <p>During time $t$, all the drops that are in a cylinder of height $ut$ and base area $S_1$ equal to the area of the central cross-section of the ball perpendicular to $u$ fall on a stationary ball. That is, in the first case</p> |
| <p class="exp">$$N_1 = utS_1$$</p> | | <p class="exp">$$N_1 = utS_1$$</p> |
| <p>In the second case, we move to the reference frame of a rolling ball, then the drops fall on it with a speed of </p> | | <p>In the second case, we move to the reference frame of a rolling ball, then the drops fall on it with a speed of </p> |
| <p class="exp">$$u_1 = \sqrt{u^2 + v^2}$$</p> | | <p class="exp">$$u_1 = \sqrt{u^2 + v^2}$$</p> |
| <p class="exp">$$N_2 = u_1 t S_2$$</p> | | <p class="exp">$$N_2 = u_1 t S_2$$</p> |
| <p>where $S_2$ is equal to the area of the central cross-section of the ball perpendicular to $u_1$. For a ball-ball $S_1 = S_2 = S$, and then</p> | | <p>where $S_2$ is equal to the area of the central cross-section of the ball perpendicular to $u_1$. For a ball-ball $S_1 = S_2 = S$, and then</p> |
| <p class="exp">$$N_2 = \sqrt{u^2 + v^2} t S$$</p> | | <p class="exp">$$N_2 = \sqrt{u^2 + v^2} t S$$</p> |
| <p>Where we get the desired ratio</p> | | <p>Where we get the desired ratio</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ \fbox{$\dfrac{N_2}{N_1} = \sqrt{1 + \dfrac{v^2}{u^2}}$} $$ | | $$ \fbox{$\dfrac{N_2}{N_1} = \sqrt{1 + \dfrac{v^2}{u^2}}$} $$ |
| </p> | | </p> |
| <p>When the ball has a shape different from a sphere, its cross-sectional area will vary from the plane of the section </p> | | <p>When the ball has a shape different from a sphere, its cross-sectional area will vary from the plane of the section </p> |
| <p class="exp">$$S_1\neq S_2\neq S$$</p> | | <p class="exp">$$S_1\neq S_2\neq S$$</p> |
| <p>Thus, the answer obtained will also vary depending on the shape of the ball </p> | | <p>Thus, the answer obtained will also vary depending on the shape of the ball </p> |
| <p >NO: For additional reading, I recommend the discussion of this problem on <a href="https://ask.bc-pf.org/t/zadacha-pro-dozhd-i-myach-zadachnik-1001/3855" target="_blank">ask.bc-pf.org</a> </p> | | <p >NO: For additional reading, I recommend the discussion of this problem on <a href="https://ask.bc-pf.org/t/zadacha-pro-dozhd-i-myach-zadachnik-1001/3855" target="_blank">ask.bc-pf.org</a> </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| <p>More drops fall on a rolling ball in $\dfrac{N_2}{N_1} = \sqrt{1 + \dfrac{v^2}{u^2}}$ times.</p> | | <p>More drops fall on a rolling ball in $\dfrac{N_2}{N_1} = \sqrt{1 + \dfrac{v^2}{u^2}}$ times.</p> |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
