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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> |
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| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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 $u$. | | $2.1.35^*.$ 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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| For problem $2.1.35^*$ | | For problem $2.1.35^*$ |
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| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.35/sol.png" id="hot-airbaloon" | | <img src="https://savchenkosolutions.com/2/2.1.35/sol.png" id="hot-airbaloon" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Forces acting on a balloon | | Forces acting on a balloon |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| <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 $\vec{F}_{пт}$. | | 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> |
| <p> | | <p> |
| Let $\vec{v}^{ \prime}$ denote the velocity of the balloon relative to the Earth. Then | | Let $\vec{v}^{ \prime}$ denote the velocity of the balloon relative to the Earth. Then |
| $$ \vec{F}_{c} = - \alpha ( \vec{v}^{ \prime} - \vec{u}) $$ | | $$ \vec{F}_{c} = - \alpha ( \vec{v}^{ \prime} - \vec{u}) $$ |
| From the condition that the balloon moves uniformly in the horizontal direction, it follows | | From the condition that the balloon moves uniformly in the horizontal direction, it follows |
| $$ | \vec{F}_{c} | - | \vec{F}_{тр} | = 0 $$ | | $$ | \vec{F}_{c} | - | \vec{F}_{тр} | = 0 $$ |
| | | |
| $$ | \vec{F}| + | \vec{N}| - M | \vec{g} | = 0 $$ | | $$ | \vec{F}| + | \vec{N}| - M | \vec{g} | = 0 $$ |
| Besides, | | Besides, |
| $$ | \vec{F}_{тр} | = \mu | \vec{N} | $$ | | $$ | \vec{F}_{тр} | = \mu | \vec{N} | $$ |
| Taking into account that | | Taking into account that |
| $$ | \vec{F}_{c}| = - \alpha ( | \vec{v}^{ \prime} | - | \vec{u} | ) $$ | | $$ | \vec{F}_{c}| = - \alpha ( | \vec{v}^{ \prime} | - | \vec{u} | ) $$ |
| from the last three equations we get | | from the last three equations we get |
| $$ | \vec{v}^{ \prime} | = | \vec{u} | - \frac{ \mu}{ \alpha} (M | \vec{g} | - | \vec{F} | ) $$ | | $$ | \vec{v}^{ \prime} | = | \vec{u} | - \frac{ \mu}{ \alpha} (M | \vec{g} | - | \vec{F} | ) $$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $v = u − \sqrt{(\mu /\alpha )(mg − F)}$ at $\alpha u^2 \geq \mu (mg − F)$, otherwise $v = 0$ | | $v = u − \sqrt{(\mu /\alpha )(mg − F)}$ at $\alpha u^2 \geq \mu (mg − F)$, otherwise $v = 0$ |
