To determine the equivalent airspeed of the Boeing 777 flying at 8.2 kilometers altitude with a Mach number of 0.84, we need to follow these steps:
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Determine the temperature at 8.2 km altitude: The standard atmosphere model gives us a temperature at a certain altitude. At 8.2 km (which is approximately 26,900 feet), the temperature can be calculated with the lapse rate. The standard temperature at sea level (0 km) is 288.15 K, and the temperature generally decreases by about 6.5 K for each kilometer of altitude.
\[ T = 288.15 \text{ K} - 6.5 \text{ K/km} \times 8.2 \text{ km} \] \[ T = 288.15 \text{ K} - 53.3 \text{ K} = 234.85 \text{ K} \]
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Calculate the speed of sound at that temperature: The speed of sound \( a \) in air can be calculated using the formula: \[ a = 340.3 \sqrt{T} \text{ m/s} \]
First, we will convert the temperature in Kelvin to the appropriate units in the formula: \[ a = 340.3 \sqrt{234.85} \text{ m/s} \approx 340.3 \times 15.33 \text{ m/s} \approx 5227.30 \text{ m/s} \]
However, this is incorrect; a better formula to use is: \[ a = \sqrt{\gamma R T} \] where \( \gamma \) is the specific heat ratio of air (approximately 1.4), and \( R \) is the specific gas constant for air (approximately 287 J/(kg·K)).
Calculate as follows: \[ a = \sqrt{1.4 \times 287 \text{ J/(kg·K)} \times 234.85 \text{ K}} \approx \sqrt{1.4 \times 287 \times 234.85} \approx 303.78 \text{ m/s} \]
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Calculate equivalent airspeed: The equivalent airspeed (EAS) can now be calculated by multiplying the speed of sound by the Mach number: \[ V = \text{Mach} \times a \] \[ V = 0.84 \times a \text{ (in m/s)} \approx 0.84 \times 303.78 \text{ m/s} \approx 255.17 \text{ m/s} \]
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Convert it to knots: We need to convert from meters per second to knots: \[ 1 \text{ knot} = 0.514444 \text{ m/s} \] \[ V \text{ (in knots)} = \frac{255.17 \text{ m/s}}{0.514444 \text{ m/s/knot}} \approx 496.75 \text{ knots} \]
Thus, the equivalent airspeed of the Boeing 777 is approximately 497 knots.
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