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+ <meta name="description" content="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.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.">
+ <meta name="author" content="Aliaksandr Melnichenka">
+ <meta name="date" content="2023-10" scheme="YYYY-MM">
+ <meta property="og:title" content="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.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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+ <title>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.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.</title>
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+ <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="../#1.2">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <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.
+</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.
+
+</p>
+<center>
+ <figure>
+ <img src="statement.png"
+ loading="lazy" width="160" />
+ <figcaption>
+ For problem $1.2.11$
+ </figcaption>
+ </figure>
+</center>
+<p>
+ </p>
+
+ <h3>Solution</h3>
+ <p>
+
+ <center>
+ <figure>
+ <img src="drawing1.png"
+ loading="lazy" width="200" />
+ <figcaption>
+ Cone vessel
+ </figcaption>
+ </figure>
+ </center>
+
+$a)$ By time $t$, the water level will be $v_0t$. And the rate of change of volume will be equal to:
+$$\frac{dV}{dt} = \frac{\pi r(t)^2 dx}{dt}$$
+Where $dx$ is the change in water level:
+
+$$\frac{dV}{dt} = \pi r(t)^2 v_0$$
+
+From Geometry,
+$$r(t) = v_0 t \cdot tg(\alpha)$$
+
+By definition, the velocity of incoming water is equal to
+$$v = \frac{dV}{sdt}$$
+
+Substituting the previous expressions:
+$$v = {\pi v_0^3 t^2 \cdot tg^2(\alpha)}/s$$
+
+$b)$ For a small time interval $dt$, the volume changes by
+$$dV = q dt$$
+
+Also the volume increment can be written as
+$$dV = 2\pi r dr \cdot h $$
+
+Thus:
+
+$$ q dt = 2\pi r dr \cdot h$$
+
+Considering $v = \frac{dr}{dt}$,
+$$ \fbox{$v = \frac{q}{2\pi r h}$}$$
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $$v=\frac{\pi v_0^3t^2\operatorname{tg}^2\alpha}{s}$$
+$$v=\frac{1}{2}\sqrt{\frac{q}{\pi ht}}$$
+ </p>
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