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An observer at sea level does not hear an aircraft flying at an altitude of 7000 m until it is 13 km away from the observer. Estimate the Mach number and velocity of the aircraft. Assume that the average temperature of the air between sea level and 7000 m altitude is about -10°C.

Answer :

To estimate the Mach number and velocity of the aircraft, we need to understand the concept of the speed of sound, which varies with temperature, and the relation of that speed to the Mach number.

  1. Speed of Sound Calculation

    The speed of sound in air at different temperatures is given by the formula:

    [tex]c = \sqrt{\gamma \cdot R \cdot T}[/tex]

    where:

    • [tex]c[/tex] is the speed of sound in meters per second (m/s).
    • [tex]\gamma[/tex] is the adiabatic index, which for air is approximately 1.4.
    • [tex]R[/tex] is the specific gas constant for air, approximately 287 J/(kg·K).
    • [tex]T[/tex] is the temperature in Kelvin.

    Given that the average temperature is -10°C, we first convert this to Kelvin:

    [tex]T = -10 + 273.15 = 263.15 \text{ K}[/tex]

    Now, substitute in the values:

    [tex]c = \sqrt{1.4 \times 287 \times 263.15} \approx 325 \text{ m/s}[/tex]

  2. Estimating Velocity of Aircraft

    Since the observer hears the aircraft sound only after it is 13 km away, it implies the aircraft is traveling faster than the speed of sound. Otherwise, they would hear it immediately.

    The Mach number [tex]M[/tex] is defined as the ratio of the speed of the aircraft [tex]v[/tex] to the speed of sound [tex]c[/tex]:

    [tex]M = \frac{v}{c}[/tex]

    The distance from the observer to where the aircraft is heard, i.e., 13 km (13000 m), corresponds to the distance traveled by sound in the time the aircraft has traveled horizontally from when it passed overhead to when it was heard.

    [tex]7000 \text{ m overhead to 13000 \text{ m} horizontally} = \sqrt{(7000)^2 + (13000)^2}[/tex]

    [tex]\text{Distance traveled by sound} \approx \sqrt{7000^2 + 13000^2} \approx 14866 \text{ m}[/tex]

    Let [tex]t[/tex] be the time when sound takes to travel 13 km, then:

    [tex]ct = 14866 \text{ m}[/tex]

    [tex]t = \frac{14866}{325} \approx 45.8 \text{ seconds}[/tex]

Now, calculate the aircraft's horizontal velocity, which matches the time:

[tex]v = \frac{13000}{45.8} \approx 283.62 \text{ m/s}[/tex]

  1. Calculating Mach Number

    Finally, substitute [tex]v[/tex] and [tex]c[/tex] back into the Mach number formula:

    [tex]M = \frac{283.62}{325} \approx 0.87[/tex]

    It's noteworthy that a Mach number less than 1 indicates subsonic speed, which seems contradictory to the problem's premise if it was interpreted as the aircraft moving supersonically. However, based on the given data and calculations, the estimated Mach number is approximately 0.87.

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