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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
| </p> | | </p> |
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| <h3 id="back-link"><a href="../../#1.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.2">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.2.11.$ a. In a conical vessel, the water level rises at a constant rate $v_0$. How does the rate of water entering a vessel through a tube of section $s$ depend on time? At time zero, the vessel is empty. | | $1.2.11.$ a. In a conical vessel, the water level rises at a constant rate $v_0$. How does the rate of water entering a vessel through a tube of section $s$ depend on time? At time zero, the vessel is empty. |
| </p> | | </p> |
| | | |
| <p> | | <p> |
| b. A jet of oil hitting the surface of the water spreads over it in a round spot of thickness $h$. How does the speed of movement of the spot boundary depend on time, if the volume of oil $q$ enters per unit of time? At the initial time, the spot radius is zero. | | b. A jet of oil hitting the surface of the water spreads over it in a round spot of thickness $h$. How does the speed of movement of the spot boundary depend on time, if the volume of oil $q$ enters per unit of time? At the initial time, the spot radius is zero. |
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| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="160" /> | | loading="lazy" width="160" /> |
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| For problem $1.2.11$ | | For problem $1.2.11$ |
| </figcaption> | | </figcaption> |
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| <p> | | <p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| <img src="drawing1.png" | | <img src="drawing1.png" |
| loading="lazy" width="200" /> | | loading="lazy" width="200" /> |
| <figcaption> | | <figcaption> |
| Cone vessel | | Cone vessel |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
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| $a)$ By time $t$, the water level will be $v_0t$. And the rate of change of volume will be equal to: | | $a)$ By time $t$, the water level will be $v_0t$. And the rate of change of volume will be equal to: |
| $$\frac{dV}{dt} = \frac{\pi r(t)^2 dx}{dt}$$ | | $$\frac{dV}{dt} = \frac{\pi r(t)^2 dx}{dt}$$ |
| Where $dx$ is the change in water level: | | Where $dx$ is the change in water level: |
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| $$\frac{dV}{dt} = \pi r(t)^2 v_0$$ | | $$\frac{dV}{dt} = \pi r(t)^2 v_0$$ |
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| From Geometry, | | From Geometry, |
| $$r(t) = v_0 t \cdot \tan\alpha $$ | | $$r(t) = v_0 t \cdot \tan\alpha $$ |
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| By definition, the velocity of incoming water is equal to | | By definition, the velocity of incoming water is equal to |
