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| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| </p> | | </p> |
| </header> | | </header> |
| | | |
| <h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.6">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.6.41^*.$ Two probes are launched from an orbital station moving at a speed of $u$ in a circular orbit around the planet. The initial velocity of the probes relative to the planet is $v$ ($\sqrt{2} u > v > u$). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet. | | $2.6.41^*.$ Two probes are launched from an orbital station moving at a speed of $u$ in a circular orbit around the planet. The initial velocity of the probes relative to the planet is $v$ ($\sqrt{2} u > v > u$). One probe moves in the direction of the planet's radius; the initial velocity of the other probe is perpendicular to its radius. Find the ratio of the maximum possible distances from the probes to the center of the planet. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Let's consider two cases:</p><p> | | Let's consider two cases:</p><p> |
| 1) Velocity $\vec{v}$ in radial direction:</p><p> | | 1) Velocity $\vec{v}$ in radial direction:</p><p> |
| For this case, let's consider preservation of energy | | For this case, let's consider preservation of energy |
| $$\frac{mv^2}{2}-\frac{GmM}{R_0}=\frac{m{v'}^2}{2}-\frac{GmM}{R_1}$$ | | $$\frac{mv^2}{2}-\frac{GmM}{R_0}=\frac{m{v'}^2}{2}-\frac{GmM}{R_1}$$ |
| Analyzing the components, when $v'\to 0,$ $R_1$ tends to the maximum value $R_1 \to R_\text{1max}$ | | Analyzing the components, when $v'\to 0,$ $R_1$ tends to the maximum value $R_1 \to R_\text{1max}$ |
| $$\frac{mv^2}{2} - \frac{GmM}{R_0} = - \frac{GmM}{R_\text{1max}}$$ | | $$\frac{mv^2}{2} - \frac{GmM}{R_0} = - \frac{GmM}{R_\text{1max}}$$ |
| | | |
| $$\frac{1}{2}v^2 = GM \left( \frac{1}{R_0} - \frac{1}{R_\text{1max}} \right) $$ | | $$\frac{1}{2}v^2 = GM \left( \frac{1}{R_0} - \frac{1}{R_\text{1max}} \right) $$ |
| 2) The velocity $v$ is perpendicular to the radius</p><p> | | 2) The velocity $v$ is perpendicular to the radius</p><p> |
| From Kepler's first law, the orbit will be an ellipse | | From Kepler's first law, the orbit will be an ellipse |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.6.41_1.png" | | <img src="2.6.41_1.png" |
| loading="lazy" width="300" /> | | loading="lazy" width="300" /> |
| <figcaption> | | <figcaption> |
| Elliptical orbit | | Elliptical orbit |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| <i>$R_0$ - radius of circular orbit<br> | | <i>$R_0$ - radius of circular orbit<br> |
| $R_1$ - maximum orbit at radial velocity<br> | | $R_1$ - maximum orbit at radial velocity<br> |
| $R_2$ - maximum orbit at perpendicular velocity</i></p><p> | | $R_2$ - maximum orbit at perpendicular velocity</i></p><p> |
| | | |
| Similarly, the preservation of energy | | Similarly, the preservation of energy |
| $$\frac{mv^2}{2}-\frac{GmM}{R_0}=\frac{m{v'}^2}{2}-\frac{GmM}{R_\text{2max}}\quad(1)$$ | | $$\frac{mv^2}{2}-\frac{GmM}{R_0}=\frac{m{v'}^2}{2}-\frac{GmM}{R_\text{2max}}\quad(1)$$ |
| Conservation of angular momentum | | Conservation of angular momentum |
| $$mvR_0=m{v'}R_\text{2max}$$ | | $$mvR_0=m{v'}R_\text{2max}$$ |
| From where | | From where |
| $$v'=v\frac{R_0}{R_\text{2max}}\quad(2)$$ | | $$v'=v\frac{R_0}{R_\text{2max}}\quad(2)$$ |
| Substituting $(2)$ into $(1)$ | | Substituting $(2)$ into $(1)$ |
| $$\frac{v^2}{2}\left(1+\frac{R_0}{R_\text{2max}}\right) = GM \frac{1}{R_0}$$ | | $$\frac{v^2}{2}\left(1+\frac{R_0}{R_\text{2max}}\right) = GM \frac{1}{R_0}$$ |
| Now, let's consider circular motion:</p><p> | | Now, let's consider circular motion:</p><p> |
| Given the centripetal acceleration $a=\frac{u^2}{R_0}$, the presence of rotational motion of which causes centrifugal force, compensating gravitational force, in the equilibrium condition. | | Given the centripetal acceleration $a=\frac{u^2}{R_0}$, the presence of rotational motion of which causes centrifugal force, compensating gravitational force, in the equilibrium condition. |
| $$\frac{mu^2}{R_0}=\frac{GmM}{R_0^2} \Leftrightarrow \frac{GM}{R_0}=u^2$$ | | $$\frac{mu^2}{R_0}=\frac{GmM}{R_0^2} \Leftrightarrow \frac{GM}{R_0}=u^2$$ |
| Substituting into the final formula of motion | | Substituting into the final formula of motion |
| $$\frac{1}{2}v^2=u^2\left(1-\frac{R_0}{R_\text{1max}}\right)$$ | | $$\frac{1}{2}v^2=u^2\left(1-\frac{R_0}{R_\text{1max}}\right)$$ |
| $$\frac{1}{2}v^2\left(1+\frac{R_0}{R_\text{2max}}\right)=u^2$$ | | $$\frac{1}{2}v^2\left(1+\frac{R_0}{R_\text{2max}}\right)=u^2$$ |
| Transforming the obtained expressions | | Transforming the obtained expressions |
| $$R_0=R_\text{1max}\cdot\left(1-\frac{1}{2}\frac{v^2}{u^2}\right)=R_\text{2max}\cdot\left(2\frac{u^2}{v^2}-1\right)$$ | | $$R_0=R_\text{1max}\cdot\left(1-\frac{1}{2}\frac{v^2}{u^2}\right)=R_\text{2max}\cdot\left(2\frac{u^2}{v^2}-1\right)$$ |
| $$R_\text{1max}\cdot \frac{2u^2-v^2}{2u^2} = R_\text{2max}\cdot \frac{2u^2-v^2}{v^2}$$ | | $$R_\text{1max}\cdot \frac{2u^2-v^2}{2u^2} = R_\text{2max}\cdot \frac{2u^2-v^2}{v^2}$$ |
| From where we express the ratio of the maximum possible distances from the probes to the center of the planet | | From where we express the ratio of the maximum possible distances from the probes to the center of the planet |
| $$\boxed{\frac{R_\text{1max}}{R_\text{2max}}=\frac{2u^2}{v^2}}$$ | | $$\boxed{\frac{R_\text{1max}}{R_\text{2max}}=\frac{2u^2}{v^2}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$R_1/R_2=2u^2/v^2$$ | | $$R_1/R_2=2u^2/v^2$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
| </p> | | </p> |
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