Translated 2.1.13-2.1.33
en/2.1.28.md
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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <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?"> | ||
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| + | <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?"> | ||
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| + | <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> | ||
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| + | <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="../../#2.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <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? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.28/sol.png" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | Forces acting on the car | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <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 | ||
| + | $$ ma = \mu mg $$ | ||
| + | |||
| + | $$ a = \mu g $$ | ||
| + | 2. The kinematic equations of motion in this case are represented as follows: | ||
| + | $$ \left\{\begin{matrix} | ||
| + | v = v_0-at \\ | ||
| + | x=v_0-\frac{at^2}{2} | ||
| + | \end{matrix}\right. $$ | ||
| + | |||
| + | $$ t=\frac{v_0}{a}=\frac{v_0}{\mu g} $$ | ||
| + | 3. When substituting the values of acceleration and time into the second equation, we obtain the equation for the braking distance of a car | ||
| + | $$ x=\frac{v_0^2}{\mu g}-\frac{v_0^2}{2\mu g}=\frac{v_0^2}{2\mu g} $$ | ||
| + | |||
| + | $$ v_0 = \sqrt{2\mu gx} $$ | ||
| + | Hence the speed needs to be | ||
| + | reduced by $\sqrt{10}$ times | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer: $\sqrt{10}$ times</h4> | ||
| + | |||
| + | |||
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| + | <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> alex@savchenkosolutions.com <br></small> | ||
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| @@ -0,0 +1,99 @@ | |||
| <!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="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="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: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 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> | |||
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | |||
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| }, | |||
| 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="../../#2.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <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? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.28/sol.png" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| Forces acting on the car | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <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 | |||
| $$ ma = \mu mg $$ | |||
| $$ a = \mu g $$ | |||
| 2. The kinematic equations of motion in this case are represented as follows: | |||
| $$ \left\{\begin{matrix} | |||
| v = v_0-at \\ | |||
| x=v_0-\frac{at^2}{2} | |||
| \end{matrix}\right. $$ | |||
| $$ t=\frac{v_0}{a}=\frac{v_0}{\mu g} $$ | |||
| 3. When substituting the values of acceleration and time into the second equation, we obtain the equation for the braking distance of a car | |||
| $$ x=\frac{v_0^2}{\mu g}-\frac{v_0^2}{2\mu g}=\frac{v_0^2}{2\mu g} $$ | |||
| $$ v_0 = \sqrt{2\mu gx} $$ | |||
| Hence the speed needs to be | |||
| reduced by $\sqrt{10}$ times | |||
| </p> | |||
| </p> | |||
| <h4>Answer: $\sqrt{10}$ times</h4> | |||
| <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> alex@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||