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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> |
| </p> | | </p> |
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| <h3 id="back-link"><a href="../../#1.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.1.11^*.$ A supersonic airplane is flying horizontally. Two microphones on the same vertical at a distance $l$ from each other register the arrival of sound from an airplane flying over the microphones with a time lag $\Delta t$. The speed of sound in air is $c$. What is the speed of the plane? | | $1.1.11^*.$ A supersonic airplane is flying horizontally. Two microphones on the same vertical at a distance $l$ from each other register the arrival of sound from an airplane flying over the microphones with a time lag $\Delta t$. The speed of sound in air is $c$. What is the speed of the plane? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="1.1.11.png" | | <img src="1.1.11.png" |
| loading="lazy" alt="1.1.11" width="250" /> | | loading="lazy" alt="1.1.11" width="250" /> |
| </figure> | | </figure> |
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| <p> | | <p> |
| 1. The more the aircraft exceeds the speed of sound, the smaller the value of $AD$, T.e. $v \sim 1/AD$. | | 1. The more the aircraft exceeds the speed of sound, the smaller the value of $AD$, T.e. $v \sim 1/AD$. |
| </p> | | </p> |
| <p> | | <p> |
| 2. If the aeroplane is travelling at sonic speed $v = s$, then $AD = AB = L$, in which case triangle | | 2. If the aeroplane is travelling at sonic speed $v = s$, then $AD = AB = L$, in which case triangle |
| $ADB$ is equilateral, or | | $ADB$ is equilateral, or |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$v = c \frac{AB}{AD} = c \frac{L}{AD} \; (1)$$ | | $$v = c \frac{AB}{AD} = c \frac{L}{AD} \; (1)$$ |
| </p> | | </p> |
| <p> | | <p> |
| 3. Let's define the elements of the triangle $ADB$ | | 3. Let's define the elements of the triangle $ADB$ |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$DB = c \cdot \Delta t\text{; }AD = \sqrt{L^2 - DB^2} \; (2)$$ | | $$DB = c \cdot \Delta t\text{; }AD = \sqrt{L^2 - DB^2} \; (2)$$ |
| </p> | | </p> |
| <p> | | <p> |
| 4. Substituting $(2)$ into equation $(1)$, we obtain | | 4. Substituting $(2)$ into equation $(1)$, we obtain |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $${v}=\frac{{cL}}{\sqrt{{L}^{2}-{c}^{2}\Delta{t}^{2}}}.$$ | | $${v}=\frac{{cL}}{\sqrt{{L}^{2}-{c}^{2}\Delta{t}^{2}}}.$$ |
| </p> | | </p> |
| | | |
| <p> | | <p> |
| <i>NO: This problem is related to the Mach wave phenomenon. This phenomenon was written about in the journal ‘Quantum’. <a href="http://kvant.mccme.ru/pdf/2010/2010-03.pdf" target="_blank">2010-03.pdf</a> </i>(42 стр.) | | <i>NO: This problem is related to the Mach wave phenomenon. This phenomenon was written about in the journal ‘Quantum’. <a href="http://kvant.mccme.ru/pdf/2010/2010-03.pdf" target="_blank">2010-03.pdf</a> </i>(42 стр.) |
| </p> | | </p> |
| </p> | | </p> |
