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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 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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<meta property="og: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.">
<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>
$1.2.11.$ 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.
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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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<center>
<figure>
<img src="statement.png"
loading="lazy" width="160" />
<figcaption>
For problem $1.2.11$
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</center>
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<h3>Solution</h3>
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<center>
<figure>
<img src="drawing1.png"
loading="lazy" width="200" />
<figcaption>
Cone vessel
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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:
$$\frac{dV}{dt} = \frac{\pi r(t)^2 dx}{dt}$$
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
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<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="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 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.">
<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.">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og: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 property="og: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.">
<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>
<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>
$1.2.11.$ 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.
</p>
</p>
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" width="160" />
loading="lazy" width="160" />
<figcaption>
<figcaption>
For problem $1.2.11$
For problem $1.2.11$
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
<p>
<p>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
<center>
<center>
<figure>
<figure>
<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>
$a)$ 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:
$$\frac{dV}{dt} = \pi r(t)^2 v_0$$
$$\frac{dV}{dt} = \pi r(t)^2 v_0$$
From Geometry,
From Geometry,
$$r(t) = v_0 t \cdot tg(\alpha)$$
$$r(t) = v_0 t \cdot tg(\alpha)$$
By definition, the velocity of incoming water is equal to
By definition, the velocity of incoming water is equal to
$$v = \frac{dV}{sdt}$$
$$v = \frac{dV}{sdt}$$
Substituting the previous expressions:
Substituting the previous expressions:
$$v = {\pi v_0^3 t^2 \cdot tg^2(\alpha)}/s$$
$$v = {\pi v_0^3 t^2 \cdot tg^2(\alpha)}/s$$
$b)$ For a small time interval $dt$, the volume changes by
$b)$ For a small time interval $dt$, the volume changes by
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
