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| <title>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.</title> | | <title>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.</title> |
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| <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> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| loading="lazy" width="320" /> | | loading="lazy" width="320" /> |
| <figcaption> | | <figcaption> |
| Jet engine demonstration | | Jet engine demonstration |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
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| <p> | | <p> |
| | | |
| \begin{equation*} | | \begin{equation*} |
| \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> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="2.2.37.jpg" | | <img src="2.2.37.jpg" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Reference frame relative to the earth | | Reference frame relative to the earth |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
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| | | |
| From the condition: | | From the condition: |
| $$ | | $$ |
| u_{\text{rel}} = u, \quad v_{\text{rel}} = v | | 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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