| <!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="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> | | <meta name="description" content="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> |
| <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="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> | | <meta property="og:title" content="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> | | <meta property="og:description" content="In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body.</title> | | <title>In case of an elastic impact of a body against a fixed wall, its velocity v changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed u towards the body; b) at a speed w < v in the direction of movement of the body.</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;"> |
| <div id = "logo"> | | <div id = "logo"> |
| <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> |
| $1.4.8.$ In case of an elastic impact of a body against a fixed wall, its velocity $v$ changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed $u$ towards the body; b) at a speed $w < v$ in the direction of movement of the body. | | $1.4.8.$ In case of an elastic impact of a body against a fixed wall, its velocity $v$ changes only in the direction. Determine the change after the impact of the velocity of this body, if the wall is moving: a) at a speed $u$ towards the body; b) at a speed $w < v$ in the direction of movement of the body. |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.4.8/statement.png" | | <img src="https://savchenkosolutions.com/1/1.4.8/statement.png" |
| loading="lazy" width="180" /> | | loading="lazy" width="180" /> |
| <figcaption> | | <figcaption> |
| For problem $1.4.8$ | | For problem $1.4.8$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>The whole point of the problem is to move to an inertial system associated with the wall, correctly describing the relative velocity.</p> | | <p>The whole point of the problem is to move to an inertial system associated with the wall, correctly describing the relative velocity.</p> |
| <p>$a)$ In the case when the wall "catches up" with the ball, it is necessary to move to a frame of reference in which the wall is at rest.</p> | | <p>$a)$ In the case when the wall "catches up" with the ball, it is necessary to move to a frame of reference in which the wall is at rest.</p> |
| <p>In this frame of reference, the relative velocity of the ball before the impact is</p> | | <p>In this frame of reference, the relative velocity of the ball before the impact is</p> |
| <p class="exp">$$v_{0} + u$$</p> | | <p class="exp">$$v_{0} + u$$</p> |
| <p>After the elastic impact, the velocity projection changes sign: it becomes equal to </p> | | <p>After the elastic impact, the velocity projection changes sign: it becomes equal to </p> |
| <p class="exp">$$(- v_{0} - u)$$</p> | | <p class="exp">$$(- v_{0} - u)$$</p> |
| <p>If we then return to the original (laboratory) frame of reference, where the wall moves with a velocity of $u$, then the velocity of the ball here is </p> | | <p>If we then return to the original (laboratory) frame of reference, where the wall moves with a velocity of $u$, then the velocity of the ball here is </p> |
| <p class="exp"> | | <p class="exp"> |
| $${v}' = - v_{0} - 2u$$</p> | | $${v}' = - v_{0} - 2u$$</p> |
| <p>The change in velocity in this case is </p> | | <p>The change in velocity in this case is </p> |
| <p class="exp"> | | <p class="exp"> |
| $$\fbox{$\Delta v = {v}' - v_{0}$ $= - 2(v_{0} + u)$}$$ | | $$\fbox{$\Delta v = {v}' - v_{0}$ $= - 2(v_{0} + u)$}$$ |
| </p> | | </p> |
| <p>$b)$ Similarly, we need to go to the frame of reference in which the wall is at rest. </p> | | <p>$b)$ Similarly, we need to go to the frame of reference in which the wall is at rest. </p> |
| <p>In this frame of reference, the relative velocity of the ball before the impact is $v_{0} - w$. After the elastic impact, the projection of the velocity changes sign: it becomes equal to </p> | | <p>In this frame of reference, the relative velocity of the ball before the impact is $v_{0} - w$. After the elastic impact, the projection of the velocity changes sign: it becomes equal to </p> |
| <p class="exp"> | | <p class="exp"> |
| $$(- v_{0} + w)$$ | | $$(- v_{0} + w)$$ |
| </p> | | </p> |
| <p>If we then return to the original (laboratory) frame of reference, where the wall moves with the velocity $w$, then the velocity of the ball is </p> | | <p>If we then return to the original (laboratory) frame of reference, where the wall moves with the velocity $w$, then the velocity of the ball is </p> |
| <p class="exp"> | | <p class="exp"> |
| $${v}' = - v_{0} + 2w$$ | | $${v}' = - v_{0} + 2w$$ |
| </p> | | </p> |
| <p>The change in velocity in this case is </p> | | <p>The change in velocity in this case is </p> |
| <p class="exp"> | | <p class="exp"> |
| $$\fbox{$\Delta v = {v}' - v_{0}$ $= - 2(v_{0} - w)$}$$ | | $$\fbox{$\Delta v = {v}' - v_{0}$ $= - 2(v_{0} - w)$}$$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $\begin{aligned}&{a) }\Delta v = -2(v + u).\quad б) \Delta v = -2(v - w). (\text{Projection on the direction of the initial}\\&\text{speeds are considered positive.)}\end{aligned}$ | | $\begin{aligned}&{a) }\Delta v = -2(v + u).\quad б) \Delta v = -2(v - w). (\text{Projection on the direction of the initial}\\&\text{speeds are considered positive.)}\end{aligned}$ |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> |
| </p> | | </p> |
| <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> | | <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> | | </p> |
| </footer> | | </footer> |
| </body> | | </body> |
| | | |
| </html> | | </html> |
