Updated en 2.2.37, 2.2.44, 2.2.36, 2.2.39

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

			<h3 id="back-link"><a href="../#2.2">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>
			<p>
			$2.2.37.$ Determine the thrust force of an air-jet engine of an airplane flying at a speed of $V$. The mass consumption of fuel and air entering the engine is equal to $\mu_1$ and $\mu_2$, respectively. Velocity of combustion products relative to the aircraft at the engine exit $u$.
			</p>

		@@ -50,13 +50,14 @@
			<h3>Solution</h3>
			<p>

		+
			</p>
			<center>
			<figure>
		−	<img src="2.2.37~~.jp~~g"
		−	loading="lazy" width="250" />
		+	<img src="2.2.37_suply.png"
		+	loading="lazy" width="320" />
			<figcaption>
		−	~~Reference frame relative to the earth~~
		+	Jet engine demonstration
			</figcaption>
			</figure>
			</center>
			<p>

			\begin{equation*}
		@@ -66,9 +67,22 @@
			\Delta p_x = ((\mu_1 + \mu_2)u - \mu_2 v) \Delta t
			\end{equation*}

		+	</p>
		+	<center>
		+	<figure>
		+	<img src="2.2.37.jpg"
		+	loading="lazy" width="250" />
		+	<figcaption>
		+	Reference frame relative to the earth
		+	</figcaption>
		+	</figure>
		+	</center>
		+	<p>
		+
		+
			From the condition:
			$$
		−	u_{\text{rel}} = u, \quad v_{\text{rel}} = 0
		+	u_{\text{rel}} = u, \quad v_{\text{rel}} = v
			$$

			Write the vector sum for the relative velocities
			$$
			\vec{v}_{\text{rel}} = \vec{0} - \vec{v}
			$$

			$$
			v_{\text{rel} \, x} = -v_x
			$$

			$$
			+v_{\text{rel}} = -(-v) = v
			$$

			Find the change in momentum over a soapy time interval

			\begin{equation*}
			\Delta p_x = \left( \mu_1 u + \mu_2 (u - v) \right) \Delta t
			\end{equation*}

			Let's use Newton's second law in impulse form

			\begin{equation*}
			F_x = \frac{\Delta p_x}{\Delta t} = \mu_1 u + \mu_2 (u - v)
			\end{equation*}

			By definition of mass consumption

			$$
			m_1 = \mu_1 \Delta t, \quad m_2 = \mu_2 \Delta t
			$$

			According to Newton's third law, these forces are equal to

			\begin{equation*}
			T = -F_x
			\end{equation*}
			From where the required thrust force
			$$
			\boxed{\|T\| = \mu_1 u + \mu_2 (u - v)}
			$$
			</p>

			<h4>Answer</h4>
			<p>
			$$F = \mu_2 (u - v) + \mu_1 u$$
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			Almaskhan Arsen<br>
			</p>


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			<small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small>
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	<body style="">		<body style="">
	<header style="text-align:center;">		<header style="text-align:center;">
	<h2>Solutions of Savchenko Physics Textbook</h2>		<h2>Solutions of Savchenko Physics Textbook</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="../#2.2">$\leftarrow$Back</a></h3>		<h3 id="back-link"><a href="../#2.2">$\leftarrow$Back</a></h3>

	<h3> Statement </h3>		<h3> Statement </h3>
	<p>		<p>
	$2.2.37.$ Determine the thrust force of an air-jet engine of an airplane flying at a speed of $V$. The mass consumption of fuel and air entering the engine is equal to $\mu_1$ and $\mu_2$, respectively. Velocity of combustion products relative to the aircraft at the engine exit $u$.		$2.2.37.$ Determine the thrust force of an air-jet engine of an airplane flying at a speed of $V$. The mass consumption of fuel and air entering the engine is equal to $\mu_1$ and $\mu_2$, respectively. Velocity of combustion products relative to the aircraft at the engine exit $u$.
	</p>		</p>

	@@ -50,13 +50,14 @@
	<h3>Solution</h3>		<h3>Solution</h3>
	<p>		<p>


	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="2.2.37~~.jp~~g"		<img src="2.2.37_suply.png"
	loading="lazy" width="250" />		loading="lazy" width="320" />
	<figcaption>		<figcaption>
	~~Reference frame relative to the earth~~		Jet engine demonstration
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>

	\begin{equation*}		\begin{equation*}
	@@ -66,9 +67,22 @@
	\Delta p_x = ((\mu_1 + \mu_2)u - \mu_2 v) \Delta t		\Delta p_x = ((\mu_1 + \mu_2)u - \mu_2 v) \Delta t
	\end{equation*}		\end{equation*}

			</p>
			<center>
			<figure>
			<img src="2.2.37.jpg"
			loading="lazy" width="250" />
			<figcaption>
			Reference frame relative to the earth
			</figcaption>
			</figure>
			</center>
			<p>


	From the condition:		From the condition:
	$$		$$
	u_{\text{rel}} = u, \quad v_{\text{rel}} = 0		u_{\text{rel}} = u, \quad v_{\text{rel}} = v
	$$		$$

	Write the vector sum for the relative velocities		Write the vector sum for the relative velocities
	$$		$$
	\vec{v}_{\text{rel}} = \vec{0} - \vec{v}		\vec{v}_{\text{rel}} = \vec{0} - \vec{v}
	$$		$$

	$$		$$
	v_{\text{rel} \, x} = -v_x		v_{\text{rel} \, x} = -v_x
	$$		$$

	$$		$$
	+v_{\text{rel}} = -(-v) = v		+v_{\text{rel}} = -(-v) = v
	$$		$$

	Find the change in momentum over a soapy time interval		Find the change in momentum over a soapy time interval

	\begin{equation*}		\begin{equation*}
	\Delta p_x = \left( \mu_1 u + \mu_2 (u - v) \right) \Delta t		\Delta p_x = \left( \mu_1 u + \mu_2 (u - v) \right) \Delta t
	\end{equation*}		\end{equation*}

	Let's use Newton's second law in impulse form		Let's use Newton's second law in impulse form

	\begin{equation*}		\begin{equation*}
	F_x = \frac{\Delta p_x}{\Delta t} = \mu_1 u + \mu_2 (u - v)		F_x = \frac{\Delta p_x}{\Delta t} = \mu_1 u + \mu_2 (u - v)
	\end{equation*}		\end{equation*}

	By definition of mass consumption		By definition of mass consumption

	$$		$$
	m_1 = \mu_1 \Delta t, \quad m_2 = \mu_2 \Delta t		m_1 = \mu_1 \Delta t, \quad m_2 = \mu_2 \Delta t
	$$		$$

	According to Newton's third law, these forces are equal to		According to Newton's third law, these forces are equal to

	\begin{equation*}		\begin{equation*}
	T = -F_x		T = -F_x
	\end{equation*}		\end{equation*}
	From where the required thrust force		From where the required thrust force
	$$		$$
	\boxed{\|T\| = \mu_1 u + \mu_2 (u - v)}		\boxed{\|T\| = \mu_1 u + \mu_2 (u - v)}
	$$		$$
	</p>		</p>

	<h4>Answer</h4>		<h4>Answer</h4>
	<p>		<p>
	$$F = \mu_2 (u - v) + \mu_1 u$$		$$F = \mu_2 (u - v) + \mu_1 u$$
	</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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	<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>		<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small>
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