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.

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

    <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>

    <h3>Solution</h3>
    <p>
        

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

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

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
$$\boxed{|T| = \mu_1 u + \mu_2 (u - v)}$$

    <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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