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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Solutions of Savchenko Problems in Physics
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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 $u$.

    <h3>Solution</h3>
    <p>

There are five forces acting on the balloon (see fig.): gravity $\vec{F}_{т} = M \vec{g}$, buoyancy $\vec{F}$, air resistance $\vec{F}_{c}$, Earth reaction force $\vec{N}$ and Earth friction force $\vec{F}_{пт}$.

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
$$| \vec{v}^{ \prime} | = | \vec{u} | - \frac{ \mu}{ \alpha} (M | \vec{g} | - | \vec{F} | )$$

    <h4>Answer</h4>
    <p>
        $v = u − \sqrt{(\mu /\alpha )(mg − F)}$ at $\alpha u^2 \geq \mu (mg − F)$, otherwise $v = 0$
    </p>


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