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| <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="../../#1.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.2">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.2.8^*.$ The particle, after leaving the source, flies at a constant speed for a distance $L$, and then decelerates with acceleration $a$. At what speed will the particle have the shortest travel time from its departure to its stop? | | $1.2.8^*.$ The particle, after leaving the source, flies at a constant speed for a distance $L$, and then decelerates with acceleration $a$. At what speed will the particle have the shortest travel time from its departure to its stop? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p> | | <p> |
| On a constant velocity interval, the velocity is: | | On a constant velocity interval, the velocity is: |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$v_0 = \frac{L}{t}$$ | | $$v_0 = \frac{L}{t}$$ |
| </p> | | </p> |
| <p> | | <p> |
| During the deceleration interval the velocity changes as | | During the deceleration interval the velocity changes as |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $v(t) = v_0-at \Rightarrow v_0=at$ | | $v(t) = v_0-at \Rightarrow v_0=at$ |
| </p> | | </p> |
| <p> | | <p> |
| Equating the equations, we obtain | | Equating the equations, we obtain |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $L=at^2$ | | $L=at^2$ |
| </p> | | </p> |
| <p> | | <p> |
| From where | | From where |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $v_0 = \sqrt{La}$ | | $v_0 = \sqrt{La}$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$v = \sqrt{La}$$ | | $$v = \sqrt{La}$$ |
| </p> | | </p> |
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