The speed of sound in air (in m/s) depends on the temperature according to the a

Question

The speed of sound in air (in m/s) depends on the temperature according to the approximate expression below, where TC is the Celsius temperature. In dry air the temperature decreases about 1°C for every 150 m rise in altitude.

v = 331.5 + 0.607TC

(a) Assuming this change is constant up to an altitude of 9000 m, how long will it take the sound from an airplane at 4100 m to reach the ground on a day when the ground temperature is 30°C?
s

(b) Compare this to the time interval required if the air were constant at 30°C. Which time interval is longer?

air at changing temperature or air at constant temperature

Explanation / Answer

a) at 4100 m, the sound speed v = 331.5 - 0.607*(30-4100/150) = 331.5 - 18.2 = 334.7 m/s

(here we considered the formula v= 331.5 + .607 (Tc- d/150))

Sound speed in the ground is 331.5+.607*30 = 349.7 m/s

The average speed of sound for a sound wave from plane to ground, when Tground - 30 C, is
(334.7 + 349.7)/2 = 342.2 m/s.

hence the time required to reach sound at ground 4100/342.2 m/s= 12.01 s

b) in this case the speed will be contant. and this is = 331.5 + 30*.607 = 349.7 m/s

Sound will take less time in this situation.

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