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?

Solutions of Savchenko Problems in Physics

Aliaksandr Melnichenka
October 2023

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    <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?
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    <h3>Solution</h3>
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          <p>
            1. The more the aircraft exceeds the speed of sound, the smaller the value of $AD$, T.e. $v \sim 1/AD$.
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          <p>
            2. If the aeroplane is travelling at sonic speed $v = s$, then $AD = AB = L$, in which case triangle

$ADB$ is equilateral, or

$$v = c \frac{AB}{AD} = c \frac{L}{AD} \\; (1)$$

3. Let's define the elements of the triangle $ADB$

$$DB = c \cdot \Delta t\text{; }AD = \sqrt{L^2 - DB^2} \\; (2)$$

4. Substituting $(2)$ into equation $(1)$, we obtain

$${v}=\frac{{cL}}{\sqrt{{L}^{2}-{c}^{2}\Delta{t}^{2}}}.$$

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

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