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| <h2>Solutions of Savchenko Physics Textbook</h2> | | <h2>Solutions of Savchenko Physics Textbook</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="../#2.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../#2.3">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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: $l = \alpha p$. | | $2.3.35.$ 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> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
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| <center> | | <center> |
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| <img src="2.3.35.jpg" | | <img src="2.3.35.jpg" |
| loading="lazy" width="270" /> | | loading="lazy" width="270" /> |
| <figcaption> | | <figcaption> |
| | | |
| </figcaption> | | </figcaption> |
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| <p> | | <p> |
| The work done by the force $F$ when the particle moves $dl$ | | The work done by the force $F$ when the particle moves $dl$ |
| $$F dl = dA \quad (1)$$ | | $$F dl = dA \quad (1)$$ |
| From the law of conservation of energy | | From the law of conservation of energy |
| $$ \quad \frac{m u^2}{2} = \frac{p^2}{2m}$$ | | $$ \quad \frac{m u^2}{2} = \frac{p^2}{2m}$$ |
| Considering a small increment of energy | | Considering a small increment of energy |
| $$dE = \frac{d(p^2)}{2m} = \frac{p dp}{m} \quad (2)$$ | | $$dE = \frac{d(p^2)}{2m} = \frac{p dp}{m} \quad (2)$$ |
| $$ \quad dA = dE \Rightarrow F dl = \frac{p dp}{m} \\$$ | | $$ \quad dA = dE \Rightarrow F dl = \frac{p dp}{m} \\$$ |
| $$ \quad F = \frac{p}{m} \frac{dp}{dl} = \frac{p}{m} \frac{1}{\alpha} = \frac{1}{m} \frac{l}{\alpha} \frac{1}{\alpha} = \frac{l}{m \alpha^2} \quad (*)$$ | | $$ \quad F = \frac{p}{m} \frac{dp}{dl} = \frac{p}{m} \frac{1}{\alpha} = \frac{1}{m} \frac{l}{\alpha} \frac{1}{\alpha} = \frac{l}{m \alpha^2} \quad (*)$$ |
| From where | | From where |
| $$ \quad \frac{dp}{dl} = \frac{1}{\alpha} , \quad p = \frac{l}{\alpha} \\$$ | | $$ \quad \frac{dp}{dl} = \frac{1}{\alpha} , \quad p = \frac{l}{\alpha} \\$$ |
| $$ \quad \boxed{F = \frac{l}{m \alpha^2}}$$ | | $$ \quad \boxed{F = \frac{l}{m \alpha^2}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
