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en/2.1.38.md
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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
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| + | <title>The air resistance force acting on fog drops is proportional to the product of the radius and velocity: f = \gamma rv. Drops of radius r = 0.1 mm, falling from a great height, have a speed of about 1 \frac{m}{s} near the ground. What speed will drops have if their radius is half as large? ten times less?</title> | ||
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| + | <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | ||
| + | </div> | ||
| + | <p class="author"> | ||
| + | Solutions of Savchenko Problems in Physics <br> | ||
| + | <i><b>knowledge must be free</b></i> | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $2.1.38.$ The air resistance force acting on fog drops is proportional to the product of the radius and velocity: $f = \gamma rv$. Drops of radius $r = 0.1$ mm, falling from a great height, have a speed of about $1$ $\frac{m}{s}$ near the ground. What speed will drops have if their radius is half as large? ten times less? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/2/2.1.38/sol.png" | ||
| + | loading="lazy" width="80" /> | ||
| + | <figcaption> | ||
| + | Forces acting on a drop | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <p> | ||
| + | A falling drop is acted upon by two forces: the constant force of gravity, accelerating the drop's movement, and the force of air resistance, slowing its movement and increasing with the drop's speed. The force of air resistance increases until it becomes equal to the force of gravity. Then the speed stops changing, and the drop falls at a constant speed. | ||
| + | </p> | ||
| + | |||
| + | <p> | ||
| + | Let's write the equation after a long period of time: | ||
| + | $$ mg=γrv\;(1) $$ | ||
| + | Let's find $m$ through the volume $V$: | ||
| + | $$ m=ρV=\frac{4}{3} ρ \pi r^3 $$ | ||
| + | And we substitute into $(1)$: | ||
| + | $$ \frac{4}{3} ρ \pi r^3 g=γrv $$ | ||
| + | From here: | ||
| + | $$ v = \frac{4}{3} \frac{ρ \pi g}{γ} \cdot r^2 =\alpha r^2\;(2) $$ | ||
| + | |||
| + | $$ \alpha = \frac{4}{3} \frac{ρ \pi g}{γ} =\frac{v}{r^2}=10^8 \,\frac{1}{\text{m}\cdot\text{s}} $$ | ||
| + | We substitute and find the answer | ||
| + | $$ v(\frac{r}{2}) = \alpha \frac{r^2}{4}=0.25~\text{m/s} $$ | ||
| + | |||
| + | $$ v(\frac{r}{10}) = \alpha \frac{r^2}{100}=0.01~\text{m/s} $$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$v_1 ≈ 0.25 ~\text{m/s}$$ | ||
| + | $$v_2 ≈ 0.01 ~\text{m/s}$$ | ||
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| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
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| <meta name="description" content="The air resistance force acting on fog drops is proportional to the product of the radius and velocity: f = \gamma rv. Drops of radius r = 0.1 mm, falling from a great height, have a speed of about 1 \frac{m}{s} near the ground. What speed will drops have if their radius is half as large? ten times less?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="The air resistance force acting on fog drops is proportional to the product of the radius and velocity: f = \gamma rv. Drops of radius r = 0.1 mm, falling from a great height, have a speed of about 1 \frac{m}{s} near the ground. What speed will drops have if their radius is half as large? ten times less?"> | |||
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| <title>The air resistance force acting on fog drops is proportional to the product of the radius and velocity: f = \gamma rv. Drops of radius r = 0.1 mm, falling from a great height, have a speed of about 1 \frac{m}{s} near the ground. What speed will drops have if their radius is half as large? ten times less?</title> | |||
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| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <div id = "logo"> | |||
| <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span> | |||
| </div> | |||
| <p class="author"> | |||
| Solutions of Savchenko Problems in Physics <br> | |||
| <i><b>knowledge must be free</b></i> | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $2.1.38.$ The air resistance force acting on fog drops is proportional to the product of the radius and velocity: $f = \gamma rv$. Drops of radius $r = 0.1$ mm, falling from a great height, have a speed of about $1$ $\frac{m}{s}$ near the ground. What speed will drops have if their radius is half as large? ten times less? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/2/2.1.38/sol.png" | |||
| loading="lazy" width="80" /> | |||
| <figcaption> | |||
| Forces acting on a drop | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| A falling drop is acted upon by two forces: the constant force of gravity, accelerating the drop's movement, and the force of air resistance, slowing its movement and increasing with the drop's speed. The force of air resistance increases until it becomes equal to the force of gravity. Then the speed stops changing, and the drop falls at a constant speed. | |||
| </p> | |||
| <p> | |||
| Let's write the equation after a long period of time: | |||
| $$ mg=γrv\;(1) $$ | |||
| Let's find $m$ through the volume $V$: | |||
| $$ m=ρV=\frac{4}{3} ρ \pi r^3 $$ | |||
| And we substitute into $(1)$: | |||
| $$ \frac{4}{3} ρ \pi r^3 g=γrv $$ | |||
| From here: | |||
| $$ v = \frac{4}{3} \frac{ρ \pi g}{γ} \cdot r^2 =\alpha r^2\;(2) $$ | |||
| $$ \alpha = \frac{4}{3} \frac{ρ \pi g}{γ} =\frac{v}{r^2}=10^8 \,\frac{1}{\text{m}\cdot\text{s}} $$ | |||
| We substitute and find the answer | |||
| $$ v(\frac{r}{2}) = \alpha \frac{r^2}{4}=0.25~\text{m/s} $$ | |||
| $$ v(\frac{r}{10}) = \alpha \frac{r^2}{100}=0.01~\text{m/s} $$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$v_1 ≈ 0.25 ~\text{m/s}$$ | |||
| $$v_2 ≈ 0.01 ~\text{m/s}$$ | |||
| </p> | |||
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| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <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> | |||
| </p> | |||
| </footer> | |||
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