| <!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 bug flew into the room through an open window. The distance from the bug to the ceiling changedwith a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, tothe side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and theside wall of the room. Determine the speed of the bug’s flight and the position in the window throughwhich it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long."> | | <meta name="description" content="A bug flew into the room through an open window. The distance from the bug to the ceiling changedwith a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, tothe side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and theside wall of the room. Determine the speed of the bug’s flight and the position in the window throughwhich it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long."> |
| <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 bug flew into the room through an open window. The distance from the bug to the ceiling changedwith a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, tothe side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and theside wall of the room. Determine the speed of the bug’s flight and the position in the window throughwhich it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long."> | | <meta property="og:title" content="A bug flew into the room through an open window. The distance from the bug to the ceiling changedwith a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, tothe side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and theside wall of the room. Determine the speed of the bug’s flight and the position in the window throughwhich it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long."> |
| <meta property="og:image" content="img/logo.png"> | | <meta property="og:image" content="img/logo.png"> |
| <meta property="og:description" content="A bug flew into the room through an open window. The distance from the bug to the ceiling changedwith a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, tothe side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and theside wall of the room. Determine the speed of the bug’s flight and the position in the window throughwhich it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long."> | | <meta property="og:description" content="A bug flew into the room through an open window. The distance from the bug to the ceiling changedwith a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, tothe side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and theside wall of the room. Determine the speed of the bug’s flight and the position in the window throughwhich it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long."> |
| <meta name="yandex-verification" content="6cfda41f74038368"> | | <meta name="yandex-verification" content="6cfda41f74038368"> |
| <title>A bug flew into the room through an open window. The distance from the bug to the ceiling changedwith a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, tothe side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and theside wall of the room. Determine the speed of the bug’s flight and the position in the window throughwhich it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long.</title> | | <title>A bug flew into the room through an open window. The distance from the bug to the ceiling changedwith a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, tothe side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and theside wall of the room. Determine the speed of the bug’s flight and the position in the window throughwhich it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long.</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="../../#1.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.1.3.$ A bug flew into the room through an open window. The distance from the bug to the ceiling changed | | $1.1.3.$ A bug flew into the room through an open window. The distance from the bug to the ceiling changed |
| with a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, to | | with a speed of 1 m/s, the distance to the wall opposite to the window changed with a speed of 2 m/s, to |
| the side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and the | | the side wall - with a speed of 2 m/s. After 1 s of flight the bug hit the corner between the ceiling and the |
| side wall of the room. Determine the speed of the bug’s flight and the position in the window through | | side wall of the room. Determine the speed of the bug’s flight and the position in the window through |
| which it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long. | | which it flew into the room. The room is 2.5 m high, 4 m wide, and 4 m long. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p> | | <p> |
| Let the beetle hit the corner $B$. It is known that after one second of flight the beetle has hit point $B$, so the window is at a distance of $1$ m from the ceiling, since with respect to the ceiling its velocity has changed by $1$ m/s. | | Let the beetle hit the corner $B$. It is known that after one second of flight the beetle has hit point $B$, so the window is at a distance of $1$ m from the ceiling, since with respect to the ceiling its velocity has changed by $1$ m/s. |
| </p> | | </p> |
| <p> | | <p> |
| Relative to the side wall, its velocity changed by $2$ m/s. Therefore, it flew at a distance of $2$ m from the side wall. Point $A$ is the place where the beetle flew into the room. | | Relative to the side wall, its velocity changed by $2$ m/s. Therefore, it flew at a distance of $2$ m from the side wall. Point $A$ is the place where the beetle flew into the room. |
| </p> | | </p> |
| <p> | | <p> |
| The coordinates of point $A$: $1$ m from the ceiling and $2$ m from the side wall. Knowing the change of velocities along the three directions $XYZ$, let's find the flight velocity of the beetle: | | The coordinates of point $A$: $1$ m from the ceiling and $2$ m from the side wall. Knowing the change of velocities along the three directions $XYZ$, let's find the flight velocity of the beetle: |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$v = \sqrt{v_x^2 + v_y^2 + v_z^2}, \;(1)$$ | | $$v = \sqrt{v_x^2 + v_y^2 + v_z^2}, \;(1)$$ |
| </p> | | </p> |
| <p> | | <p> |
| Let's substitute the values of velocity projections into $(1)$ | | Let's substitute the values of velocity projections into $(1)$ |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$v = \sqrt{2^2 + 2^2 + 1^2} = 3 \; \frac{m}{s}$$ | | $$v = \sqrt{2^2 + 2^2 + 1^2} = 3 \; \frac{m}{s}$$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| A beetle flew into the room at point $A$ located $1$ m from the ceiling and $2$ m from the side wall with a velocity of $3\, m/s$ | | A beetle flew into the room at point $A$ located $1$ m from the ceiling and $2$ m from the side wall with a velocity of $3\, m/s$ |
| </p> | | </p> |
| | | |
| | | |
| <footer class="row container"> | | <footer class="row container"> |
| <br> | | <br> |
| <p> | | <p> |
