Translated 2.1.34-2.1.47

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+ <meta name="description" content="The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u.">
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+ <meta name="date" content="2023-10" scheme="YYYY-MM">
+ <meta property="og:title" content="The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u.">
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+ <title>The mass of the balloon together with the rope dragging along the ground is m; the buoyant force acting on the balloon is F; the coefficient of friction of the rope on the ground is \mu . The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: f = \alpha v^2. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of u.</title>
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+ <span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span>
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+ Solutions of Savchenko Problems in Physics <br>
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+ <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+ <p>
+ $2.1.35^*.$ The mass of the balloon together with the rope dragging along the ground is $m$; the buoyant force acting on the balloon is $F$; the coefficient of friction of the rope on the ground is $\mu $. The air resistance force acting on the balloon is proportional to the square of the balloon's velocity relative to the air: $f = \alpha v^2$. Find the velocity of the balloon relative to the ground if a horizontal wind is blowing with a velocity of $u$.
+
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+ <figcaption>
+ For problem $2.1.35^*$
+ </figcaption>
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+<p>
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+
+ <h3>Solution</h3>
+ <p>
+
+</p>
+<center>
+ <figure>
+ <img src="https://savchenkosolutions.com/2/2.1.35/sol.png" id="hot-airbaloon"
+ loading="lazy" width="350" />
+ <figcaption>
+ Forces acting on a balloon
+ </figcaption>
+ </figure>
+</center>
+<p>
+
+<p>
+There are five forces acting on the balloon <a href="#hot-airbaloon">(see fig.)</a>: gravity $\vec{F}_{т} = M \vec{g}$, buoyancy $\vec{F}$, air resistance $\vec{F}_{c}$, Earth reaction force $\vec{N}$ and Earth friction force $\vec{F}_{пт}$.
+</p>
+<p>
+Let $\vec{v}^{ \prime}$ denote the velocity of the balloon relative to the Earth. Then
+$$ \vec{F}_{c} = - \alpha ( \vec{v}^{ \prime} - \vec{u}) $$
+From the condition that the balloon moves uniformly in the horizontal direction, it follows
+$$ | \vec{F}_{c} | - | \vec{F}_{тр} | = 0 $$
+
+$$ | \vec{F}| + | \vec{N}| - M | \vec{g} | = 0 $$
+Besides,
+$$ | \vec{F}_{тр} | = \mu | \vec{N} | $$
+Taking into account that
+$$ | \vec{F}_{c}| = - \alpha ( | \vec{v}^{ \prime} | - | \vec{u} | ) $$
+from the last three equations we get
+$$ | \vec{v}^{ \prime} | = | \vec{u} | - \frac{ \mu}{ \alpha} (M | \vec{g} | - | \vec{F} | ) $$
+</p>
+ </p>
+
+ <h4>Answer</h4>
+ <p>
+ $v = u − \sqrt{(\mu /α)(mg − F)}$ at $αu^2 \geq \mu (mg − F)$, otherwise $v = 0$
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