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			<h2>Solutions of Savchenko Problems in Physics</h2>
			<p class="author">
			Aliaksandr Melnichenka <br/>
			October 2023
			</p>
			</header>

			<h3 id="back-link"><a href="../../#1.1">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<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?
			</p>

			<h3>Solution</h3>
			<p>
			<center>
			<figure>
			<img src="1.1.11.png"
			loading="lazy" alt="1.1.11" width="250" />
			</figure>
			</center>
			<p>
			1. The more the aircraft exceeds the speed of sound, the smaller the value of $AD$, T.e. $v \sim 1/AD$.
			</p>
			<p>
			2. If the aeroplane is travelling at sonic speed $v = s$, then $AD = AB = L$, in which case triangle
			$ADB$ is equilateral, or
			</p>
			<p style="text-align: center;">
			$$v = c \frac{AB}{AD} = c \frac{L}{AD} \; (1)$$
			</p>
			<p>
			3. Let's define the elements of the triangle $ADB$
			</p>
			<p style="text-align: center;">
			$$DB = c \cdot \Delta t\text{; }AD = \sqrt{L^2 - DB^2} \; (2)$$
			</p>
			<p>
			4. Substituting $(2)$ into equation $(1)$, we obtain
			</p>
			<p style="text-align: center;">
			$${v}=\frac{{cL}}{\sqrt{{L}^{2}-{c}^{2}\Delta{t}^{2}}}.$$
			</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.)
			</p>
			</p>

			<h4>Answer</h4>
			<p>
			$$v = cl/\sqrt{l^{2}-{c}^{2}{∆t}^{2}}$$

			</p>


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	<header style="text-align:center;">		<header style="text-align:center;">
	<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="../../#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>
	</center>		</center>
	<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>

	<h4>Answer</h4>		<h4>Answer</h4>
	<p>		<p>
	$$v = cl/\sqrt{l^{2}-{c}^{2}{∆t}^{2}}$$		$$v = cl/\sqrt{l^{2}-{c}^{2}{∆t}^{2}}$$

	</p>		</p>


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