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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | |||
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| <p class="author"> | |||
| Solutions of Savchenko Problems in Physics <br> | |||
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| <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.3.27^*.$ A spherical tank standing on the ground has a radius of $R$. What is the lowest speed at which a rock thrown from the ground can fly over the reservoir just by touching its top? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| @@ -57,12 +57,12 @@ | |||
| <p> | |||
| <p>The stone must be thrown at an angle $\alpha$ to the horizon, satisfying the equations obtained in <a href="../1.3.6" target="_blank">1.3.6</a>:</p> | |||
| <p class="exp"> | |||
| − | $$v_x = v \cos | ||
| − | $$x = vt \cos | ||
| + | $$v_x = v \cos\alpha ; \quad v_y = v \sin\alpha - gt;$$ | ||
| + | $$x = vt \cos\alpha ; \quad y = vt \sin\alpha - gt^2 / 2.$$ | ||
| </p> | |||
| <p>The time it takes for the stone to rise to the maximum height $2R$ is found as</p> | |||
| <p class="exp"> | |||
| − | $$ t_1 = \frac{v_0 \sin | ||
| + | $$ t_1 = \frac{v_0 \sin\alpha}{g} $$ | ||
| </p> | |||
| <p>The maximum height of the stone lift along the vertical axis should be equal to $y_{max} = 2R$, therefore</p> | |||
| <p class="exp"> | |||
| $$ \frac{v_0^2 \sin^2 \alpha}{2g} = 2R $$ | |||
| </p> | |||
| <p>Determine the value of the initial throw speed</p> | |||
| <p class="exp"> | |||
| $$ v_0 = \sqrt{\frac{4gR}{\sin^2 \alpha}} $$ | |||
| @@ -74,9 +74,9 @@ | |||
| </p> | |||
| <p>The angle $\alpha$ at which the stone should be thrown is determined from the initial conditions</p> | |||
| <p class="exp"> | |||
| − | $$ v_0t_1 \cos | ||
| − | $$ 2R = \frac{v_0^2 \sin | ||
| − | $$\tan | ||
| + | $$ v_0t_1 \cos\alpha = \frac{v_0^2 \sin\alpha\cos\alpha}{g}$$ | ||
| + | $$ 2R = \frac{v_0^2 \sin\alpha\cos\alpha}{g}$$ | ||
| + | $$\tan\alpha = 2; \quad \alpha = \text{\arctan } \;2 \approx 63^\circ$$ | ||
| </p> | |||
| <p>Substituting into the formula for $v_0$</p> | |||
| <p class="exp"> | |||
| $$ \fbox{$v_0 = \sqrt{\frac{4gR}{\sin^2 63^\circ}} = \sqrt{5Rg}$} $$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$v = \sqrt{5gR}$$ | |||
| </p> | |||
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| </a> | </a> | ||
| <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> | ||
| </p> | </p> | ||
| </header> | </header> | ||
| <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | ||
| <h3> Statement </h3> | <h3> Statement </h3> | ||
| <p> | <p> | ||
| $1.3.27^*.$ A spherical tank standing on the ground has a radius of $R$. What is the lowest speed at which a rock thrown from the ground can fly over the reservoir just by touching its top? | $1.3.27^*.$ A spherical tank standing on the ground has a radius of $R$. What is the lowest speed at which a rock thrown from the ground can fly over the reservoir just by touching its top? | ||
| </p> | </p> | ||
| <h3>Solution</h3> | <h3>Solution</h3> | ||
| @@ -57,12 +57,12 @@ | |||
| <p> | <p> | ||
| <p>The stone must be thrown at an angle $\alpha$ to the horizon, satisfying the equations obtained in <a href="../1.3.6" target="_blank">1.3.6</a>:</p> | <p>The stone must be thrown at an angle $\alpha$ to the horizon, satisfying the equations obtained in <a href="../1.3.6" target="_blank">1.3.6</a>:</p> | ||
| <p class="exp"> | <p class="exp"> | ||
| $$v_x = v \cos |
$$v_x = v \cos\alpha ; \quad v_y = v \sin\alpha - gt;$$ | ||
| $$x = vt \cos |
$$x = vt \cos\alpha ; \quad y = vt \sin\alpha - gt^2 / 2.$$ | ||
| </p> | </p> | ||
| <p>The time it takes for the stone to rise to the maximum height $2R$ is found as</p> | <p>The time it takes for the stone to rise to the maximum height $2R$ is found as</p> | ||
| <p class="exp"> | <p class="exp"> | ||
| $$ t_1 = \frac{v_0 \sin |
$$ t_1 = \frac{v_0 \sin\alpha}{g} $$ | ||
| </p> | </p> | ||
| <p>The maximum height of the stone lift along the vertical axis should be equal to $y_{max} = 2R$, therefore</p> | <p>The maximum height of the stone lift along the vertical axis should be equal to $y_{max} = 2R$, therefore</p> | ||
| <p class="exp"> | <p class="exp"> | ||
| $$ \frac{v_0^2 \sin^2 \alpha}{2g} = 2R $$ | $$ \frac{v_0^2 \sin^2 \alpha}{2g} = 2R $$ | ||
| </p> | </p> | ||
| <p>Determine the value of the initial throw speed</p> | <p>Determine the value of the initial throw speed</p> | ||
| <p class="exp"> | <p class="exp"> | ||
| $$ v_0 = \sqrt{\frac{4gR}{\sin^2 \alpha}} $$ | $$ v_0 = \sqrt{\frac{4gR}{\sin^2 \alpha}} $$ | ||
| @@ -74,9 +74,9 @@ | |||
| </p> | </p> | ||
| <p>The angle $\alpha$ at which the stone should be thrown is determined from the initial conditions</p> | <p>The angle $\alpha$ at which the stone should be thrown is determined from the initial conditions</p> | ||
| <p class="exp"> | <p class="exp"> | ||
| $$ v_0t_1 \cos |
$$ v_0t_1 \cos\alpha = \frac{v_0^2 \sin\alpha\cos\alpha}{g}$$ | ||
| $$ 2R = \frac{v_0^2 \sin |
$$ 2R = \frac{v_0^2 \sin\alpha\cos\alpha}{g}$$ | ||
| $$\tan |
$$\tan\alpha = 2; \quad \alpha = \text{\arctan } \;2 \approx 63^\circ$$ | ||
| </p> | </p> | ||
| <p>Substituting into the formula for $v_0$</p> | <p>Substituting into the formula for $v_0$</p> | ||
| <p class="exp"> | <p class="exp"> | ||
| $$ \fbox{$v_0 = \sqrt{\frac{4gR}{\sin^2 63^\circ}} = \sqrt{5Rg}$} $$ | $$ \fbox{$v_0 = \sqrt{\frac{4gR}{\sin^2 63^\circ}} = \sqrt{5Rg}$} $$ | ||
| </p> | </p> | ||
| </p> | </p> | ||
| <h4>Answer</h4> | <h4>Answer</h4> | ||
| <p> | <p> | ||
| $$v = \sqrt{5gR}$$ | $$v = \sqrt{5gR}$$ | ||
| </p> | </p> | ||
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