| <!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="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane."> | | <meta name="description" content="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane."> |
| <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="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane."> | | <meta property="og:title" content="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane."> | | <meta property="og:description" content="a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane.</title> | | <title>a. The velocity of point A of a rigid body is v and forms an angle 45^{\circ} with the direction of line AB. The velocity of point B of this body is u. Determine the projection of the velocity of point B on the direction AB. b. The velocities of points A and B of a rigid body are v. The velocity of point C, located in the plane of line AB and vector v, is u > v. Find the projection of the velocity of point C on the axis perpendicular to the specified plane.</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> |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.7.$ $a. $ The velocity of point $A$ of a rigid body is $v$ and forms an angle $45^{\circ}$ with the direction of line $AB$. The velocity of point $B$ of this body is $u$. Determine the projection of the velocity of point $B$ on the direction $AB$. | | $1.5.7.$ $a. $ The velocity of point $A$ of a rigid body is $v$ and forms an angle $45^{\circ}$ with the direction of line $AB$. The velocity of point $B$ of this body is $u$. Determine the projection of the velocity of point $B$ on the direction $AB$. |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.7/statement1.png" | | <img src="https://savchenkosolutions.com/1/1.5.7/statement1.png" |
| loading="lazy" width="130" /> | | loading="lazy" width="130" /> |
| <figcaption> | | <figcaption> |
| For problem 1.5.7.a | | For problem 1.5.7.a |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| b. The velocities of points $A$ and $B$ of a rigid body are $v$. The velocity of point $C$, located in the plane of line $AB$ and vector $v$, is $u > v$. Find the projection of the velocity of point $C$ on the axis perpendicular to the specified plane. | | b. The velocities of points $A$ and $B$ of a rigid body are $v$. The velocity of point $C$, located in the plane of line $AB$ and vector $v$, is $u > v$. Find the projection of the velocity of point $C$ on the axis perpendicular to the specified plane. |
| </p> | | </p> |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.7/statement2.png" | | <img src="https://savchenkosolutions.com/1/1.5.7/statement2.png" |
| loading="lazy" width="150" /> | | loading="lazy" width="150" /> |
| <figcaption> | | <figcaption> |
| For problem $1.5.7.b$ | | For problem $1.5.7.b$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>a) If the distances between two points are constant, then if we project the velocity of point $A$ onto line $AB$, then the projection of the velocity of point $B$ onto line $AB$ is exactly the same</p> | | <p>a) If the distances between two points are constant, then if we project the velocity of point $A$ onto line $AB$, then the projection of the velocity of point $B$ onto line $AB$ is exactly the same</p> |
| <p class="exp">$$\fbox{$u_{AB} = v/\sqrt{2}$}$$</p> | | <p class="exp">$$\fbox{$u_{AB} = v/\sqrt{2}$}$$</p> |
| | | |
| <p>b) The velocities of $A$ and $B$ are the same because either the body does not rotate, or its rotation occurs around line $AB$. </p> | | <p>b) The velocities of $A$ and $B$ are the same because either the body does not rotate, or its rotation occurs around line $AB$. </p> |
| <p>The fact that the velocity of point $C$ has a different value indicates that its velocity $\vec u$ is made up of the velocity $\vec v$ in the plane of the figure and $\vec u_1$ perpendicular to the plane.</a> | | <p>The fact that the velocity of point $C$ has a different value indicates that its velocity $\vec u$ is made up of the velocity $\vec v$ in the plane of the figure and $\vec u_1$ perpendicular to the plane.</a> |
| <p class="exp">$$\vec{u}_1+\vec{v} = \vec{u}$$</p> | | <p class="exp">$$\vec{u}_1+\vec{v} = \vec{u}$$</p> |
| <p>From where</a> | | <p>From where</a> |
| <p class="exp">$$\fbox{$u_1 = \sqrt{u^2 - v^2}$}$$</p> | | <p class="exp">$$\fbox{$u_1 = \sqrt{u^2 - v^2}$}$$</p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| a. $u_{AB} = v/\sqrt{2}$<br> | | a. $u_{AB} = v/\sqrt{2}$<br> |
| b. $u_1 = \sqrt{u^2 − v^2}$ | | b. $u_1 = \sqrt{u^2 − v^2}$ |
| </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> |
