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en/2.2.38.md
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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 water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$."> | ||
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| + | <meta property="og:description" content="A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$."> | ||
| + | <meta name="yandex-verification" content="6cfda41f74038368"> | ||
| + | <title>A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$.</title> | ||
| + | <link rel="stylesheet" href="https://savchenko-physics.github.io/css/css-latex/style.css"> | ||
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| + | </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="../#2.2">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.2.38.$ A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$. | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="draw.png" | ||
| + | loading="lazy" width="230" /> | ||
| + | <figcaption> | ||
| + | For problem $2.2.38$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | Since in steady-state mode remains unchanged, by Newton's first law, the thrust force is compensated by the drag force | ||
| + | $$F_\text{resistance}=F_\text{traction}$$ | ||
| + | |||
| + | During the time $dt$ the mass $dm$ of escaping water is: | ||
| + | $$dm=\rho S v\,dt$$ | ||
| + | In this case, relative to the Non-inertial Reference Frame, the water flies out with a velocity $u-v$. | ||
| + | </p><p> | ||
| + | Hence the momentum $dp$ of mass $dm$ | ||
| + | $$dp=\rho S v(u-v)\,dt$$ | ||
| + | The force of water thrust, we find through the change of momentum | ||
| + | $$F=\frac{dp}{dt}=\rho S v(u-v)$$ | ||
| + | Let's substitute and rewrite the equilibrium condition | ||
| + | $$\rho S v(u-v)=k v^2$$ | ||
| + | From where | ||
| + | $$\boxed{v=u \frac{\rho S}{\rho S+k}}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$v=\rho Su/(\rho S+k)$$ | ||
| + | </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> astrosander01@gmail.com <br></small> | ||
| + | </p> | ||
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| + | </html> | ||
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| <!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="A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$.</title> | |||
| <link rel="stylesheet" href="https://savchenko-physics.github.io/css/css-latex/style.css"> | |||
| <link rel="icon" href="https://savchenko-physics.github.io/img/logo.png" type="image/png"> | |||
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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="../#2.2">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.2.38.$ A water jet boat moves in calm water. Force of water resistance to boat movement $F = k v^2$ . The speed of the ejected water relative to the boat $u$. Determine the steady-state speed of the boat, if the cross-section of the flow of water captured by the engine is $S$, the water density is $\rho$. | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="draw.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| For problem $2.2.38$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| Since in steady-state mode remains unchanged, by Newton's first law, the thrust force is compensated by the drag force | |||
| $$F_\text{resistance}=F_\text{traction}$$ | |||
| During the time $dt$ the mass $dm$ of escaping water is: | |||
| $$dm=\rho S v\,dt$$ | |||
| In this case, relative to the Non-inertial Reference Frame, the water flies out with a velocity $u-v$. | |||
| </p><p> | |||
| Hence the momentum $dp$ of mass $dm$ | |||
| $$dp=\rho S v(u-v)\,dt$$ | |||
| The force of water thrust, we find through the change of momentum | |||
| $$F=\frac{dp}{dt}=\rho S v(u-v)$$ | |||
| Let's substitute and rewrite the equilibrium condition | |||
| $$\rho S v(u-v)=k v^2$$ | |||
| From where | |||
| $$\boxed{v=u \frac{\rho S}{\rho S+k}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$v=\rho Su/(\rho S+k)$$ | |||
| </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> astrosander01@gmail.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||