| <!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="On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?"> | | <meta name="description" content="On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?"> |
| <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="On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?"> | | <meta property="og:title" content="On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?"> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?"> | | <meta property="og:description" content="On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?"> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?</title> | | <title>On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same?</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;"> |
| <a href="../../" style="text-decoration: none;"> | | <a href="../../" style="text-decoration: none;"> |
| <div id="logo"> | | <div id="logo"> |
| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
| </div> | | </div> |
| </a> | | </a> |
| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.1.28.$ On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same? | | $2.1.28.$ On an icy section of the highway, the coefficient of friction between the wheels and the road is ten times less than on an un-icy one. How many times do you need to reduce the speed of the car so that the braking distance on an icy section of the highway remains the same? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.28/sol.png" | | <img src="https://savchenkosolutions.com/2/2.1.28/sol.png" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the car | | Forces acting on the car |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p> | | <p> |
| 1. The external force when the car is moving is the friction force $F = \mu mg$, therefore, without taking into account the resistance from the air, the dynamic equation of motion has the form | | 1. The external force when the car is moving is the friction force $F = \mu mg$, therefore, without taking into account the resistance from the air, the dynamic equation of motion has the form |
| $$ ma = \mu mg $$ | | $$ ma = \mu mg $$ |
| | | |
| $$ a = \mu g $$ | | $$ a = \mu g $$ |
| 2. The kinematic equations of motion in this case are represented as follows: | | 2. The kinematic equations of motion in this case are represented as follows: |
| $$ \left\{\begin{matrix} | | $$ \left\{\begin{matrix} |
| v = v_0-at \\ | | v = v_0-at \\ |
| x=v_0-\frac{at^2}{2} | | x=v_0-\frac{at^2}{2} |
| \end{matrix}\right. $$ | | \end{matrix}\right. $$ |
| | | |
| $$ t=\frac{v_0}{a}=\frac{v_0}{\mu g} $$ | | $$ t=\frac{v_0}{a}=\frac{v_0}{\mu g} $$ |
| 3. When substituting the values of acceleration and time into the second equation, we obtain the equation for the braking distance of a car | | 3. When substituting the values of acceleration and time into the second equation, we obtain the equation for the braking distance of a car |
| $$ x=\frac{v_0^2}{\mu g}-\frac{v_0^2}{2\mu g}=\frac{v_0^2}{2\mu g} $$ | | $$ x=\frac{v_0^2}{\mu g}-\frac{v_0^2}{2\mu g}=\frac{v_0^2}{2\mu g} $$ |
| | | |
| $$ v_0 = \sqrt{2\mu gx} $$ | | $$ v_0 = \sqrt{2\mu gx} $$ |
| Hence the speed needs to be | | Hence the speed needs to be |
| reduced by $\sqrt{10}$ times | | reduced by $\sqrt{10}$ times |
| </p> | | </p> |
| </p> | | </p> |
| | | |
