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| + | <meta name="description" content="a. Due to air resistance, raindrops fall at a constant velocity v perpendicular to the ground surface. How is it necessary to position a cylindrical bucket on a platform moving at a speed of u so that drops do not fall on its walls? b. At a wind speed of 10 \frac{m}{s}, raindrops fall at an angle of 30^\circ to the vertical. At what wind speed will the drops fall at an angle of 45^\circ?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="a. Due to air resistance, raindrops fall at a constant velocity v perpendicular to the ground surface. How is it necessary to position a cylindrical bucket on a platform moving at a speed of u so that drops do not fall on its walls? b. At a wind speed of 10 \frac{m}{s}, raindrops fall at an angle of 30^\circ to the vertical. At what wind speed will the drops fall at an angle of 45^\circ?"> | ||
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| + | <title>a. Due to air resistance, raindrops fall at a constant velocity v perpendicular to the ground surface. How is it necessary to position a cylindrical bucket on a platform moving at a speed of u so that drops do not fall on its walls? b. At a wind speed of 10 \frac{m}{s}, raindrops fall at an angle of 30^\circ to the vertical. At what wind speed will the drops fall at an angle of 45^\circ?</title> | ||
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| + | <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="../../#1.4">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.4.5.$ a. Due to air resistance, raindrops fall at a constant velocity $v$ perpendicular to the ground surface. How is it necessary to position a cylindrical bucket on a platform moving at a speed of $u$ so that drops do not fall on its walls? | ||
| + | </p><p> | ||
| + | b. At a wind speed of $10$ $\frac{m}{s}$, raindrops fall at an angle of $30^\circ$ to the vertical. At what wind speed will the drops fall at an angle of $45^\circ$? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.4.5/draw.png" | ||
| + | loading="lazy" width="200" /> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <p > | ||
| + | a) To prevent raindrops from hitting the walls of the bucket, their relative velocity must be parallel to the walls of the bucket. | ||
| + | </p> | ||
| + | <p > | ||
| + | In the inertial reference frame associated with raindrops, the angle $\alpha$ between the relative velocity and the horizontal is defined as | ||
| + | </p> | ||
| + | <p class="exp"> | ||
| + | $$\tan \varphi = u/v \; (1)$$ | ||
| + | </p> | ||
| + | <p > | ||
| + | The walls of the bucket must be at the same angle. | ||
| + | </p> | ||
| + | <p > | ||
| + | b) Considering the relationship of the angle with the vertical and horizontal $\beta = 90^{\circ} - \alpha$. We substitute numerical data into $(1)$ and solve the system: | ||
| + | </p> | ||
| + | <p class="exp"> | ||
| + | $$\left\{\begin{matrix} | ||
| + | 10 \cdot \tan 60^{\circ} = u\\ | ||
| + | v \cdot \tan 45^{\circ} = u | ||
| + | \end{matrix}\right.$$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | <p> | ||
| + | $a.$ The bucket should be tilted in the direction of the platform's movement at an angle $\varphi$ to the vertical: $\tan \varphi = u/v.$ | ||
| + | </p> | ||
| + | <p> | ||
| + | $б.$ $u = 10\sqrt{3} \, m/s$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
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| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
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| + | <p> | ||
| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> aliaksandr@savchenkosolutions.com <br></small> | ||
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| @@ -0,0 +1,108 @@ | |||
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| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="a. Due to air resistance, raindrops fall at a constant velocity v perpendicular to the ground surface. How is it necessary to position a cylindrical bucket on a platform moving at a speed of u so that drops do not fall on its walls? b. At a wind speed of 10 \frac{m}{s}, raindrops fall at an angle of 30^\circ to the vertical. At what wind speed will the drops fall at an angle of 45^\circ?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="a. Due to air resistance, raindrops fall at a constant velocity v perpendicular to the ground surface. How is it necessary to position a cylindrical bucket on a platform moving at a speed of u so that drops do not fall on its walls? b. At a wind speed of 10 \frac{m}{s}, raindrops fall at an angle of 30^\circ to the vertical. At what wind speed will the drops fall at an angle of 45^\circ?"> | |||
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| </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="../../#1.4">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.4.5.$ a. Due to air resistance, raindrops fall at a constant velocity $v$ perpendicular to the ground surface. How is it necessary to position a cylindrical bucket on a platform moving at a speed of $u$ so that drops do not fall on its walls? | |||
| </p><p> | |||
| b. At a wind speed of $10$ $\frac{m}{s}$, raindrops fall at an angle of $30^\circ$ to the vertical. At what wind speed will the drops fall at an angle of $45^\circ$? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.4.5/draw.png" | |||
| loading="lazy" width="200" /> | |||
| </figure> | |||
| </center> | |||
| <p > | |||
| a) To prevent raindrops from hitting the walls of the bucket, their relative velocity must be parallel to the walls of the bucket. | |||
| </p> | |||
| <p > | |||
| In the inertial reference frame associated with raindrops, the angle $\alpha$ between the relative velocity and the horizontal is defined as | |||
| </p> | |||
| <p class="exp"> | |||
| $$\tan \varphi = u/v \; (1)$$ | |||
| </p> | |||
| <p > | |||
| The walls of the bucket must be at the same angle. | |||
| </p> | |||
| <p > | |||
| b) Considering the relationship of the angle with the vertical and horizontal $\beta = 90^{\circ} - \alpha$. We substitute numerical data into $(1)$ and solve the system: | |||
| </p> | |||
| <p class="exp"> | |||
| $$\left\{\begin{matrix} | |||
| 10 \cdot \tan 60^{\circ} = u\\ | |||
| v \cdot \tan 45^{\circ} = u | |||
| \end{matrix}\right.$$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| <p> | |||
| $a.$ The bucket should be tilted in the direction of the platform's movement at an angle $\varphi$ to the vertical: $\tan \varphi = u/v.$ | |||
| </p> | |||
| <p> | |||
| $б.$ $u = 10\sqrt{3} \, m/s$ | |||
| </p> | |||
| </p> | |||
| <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> aliaksandr@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||