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| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Physics Textbook</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../#2.3">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.3.35.$ A particle of mass $m$ flies into a region where it is affected by a braking force that depends only on the distance between the particle and the boundary of the region. Find this dependence if the depth of penetration of a particle into the braking region is proportional to its initial momentum: $l = \alpha p$. | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="2.3.35.jpg" | ||
| + | loading="lazy" width="270" /> | ||
| + | <figcaption> | ||
| + | |||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | The work done by the force $F$ when the particle moves $dl$ | ||
| + | $$F dl = dA \quad (1)$$ | ||
| + | From the law of conservation of energy | ||
| + | $$ \quad \frac{m u^2}{2} = \frac{p^2}{2m}$$ | ||
| + | Considering a small increment of energy | ||
| + | $$dE = \frac{d(p^2)}{2m} = \frac{p dp}{m} \quad (2)$$ | ||
| + | $$ \quad dA = dE \Rightarrow F dl = \frac{p dp}{m} \\$$ | ||
| + | $$ \quad F = \frac{p}{m} \frac{dp}{dl} = \frac{p}{m} \frac{1}{\alpha} = \frac{1}{m} \frac{l}{\alpha} \frac{1}{\alpha} = \frac{l}{m \alpha^2} \quad (*)$$ | ||
| + | From where | ||
| + | $$ \quad \frac{dp}{dl} = \frac{1}{\alpha} , \quad p = \frac{l}{\alpha} \\$$ | ||
| + | $$ \quad \boxed{F = \frac{l}{m \alpha^2}}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$F={l}/{(m \alpha^2)}$$ | ||
| + | </p> | ||
| + | |||
| + | |||
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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> astrosander01@gmail.com <br></small> | ||
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| <meta property="og:description" content="A website with solutions to physics problems from Savchenko Textbook"> | |||
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| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Physics Textbook</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../#2.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.3.35.$ A particle of mass $m$ flies into a region where it is affected by a braking force that depends only on the distance between the particle and the boundary of the region. Find this dependence if the depth of penetration of a particle into the braking region is proportional to its initial momentum: $l = \alpha p$. | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="2.3.35.jpg" | |||
| loading="lazy" width="270" /> | |||
| <figcaption> | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| The work done by the force $F$ when the particle moves $dl$ | |||
| $$F dl = dA \quad (1)$$ | |||
| From the law of conservation of energy | |||
| $$ \quad \frac{m u^2}{2} = \frac{p^2}{2m}$$ | |||
| Considering a small increment of energy | |||
| $$dE = \frac{d(p^2)}{2m} = \frac{p dp}{m} \quad (2)$$ | |||
| $$ \quad dA = dE \Rightarrow F dl = \frac{p dp}{m} \\$$ | |||
| $$ \quad F = \frac{p}{m} \frac{dp}{dl} = \frac{p}{m} \frac{1}{\alpha} = \frac{1}{m} \frac{l}{\alpha} \frac{1}{\alpha} = \frac{l}{m \alpha^2} \quad (*)$$ | |||
| From where | |||
| $$ \quad \frac{dp}{dl} = \frac{1}{\alpha} , \quad p = \frac{l}{\alpha} \\$$ | |||
| $$ \quad \boxed{F = \frac{l}{m \alpha^2}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$F={l}/{(m \alpha^2)}$$ | |||
| </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> | |||