Solve Airplane Sound Wave Problem: Speed & Altitude

[the_storm]

Homework Statement

The speed of the sound in the air (in meter per second) depends on temperature according to approximate expression
V= 313.5 + 0.607T_c
where T_c is the Celsius temperature. In dry air, the temperature decreases about 1 degree Celsius for every 150M rise in altitude

Homework Equations

a) assume the change is constant up to an altitude of 9000m what time interval is required for the sound from airplane flying at 9000m to reach the ground on a day when the ground temperature is 30 Celsius

 

[aim1732]

Please show some work.

 

[the_storm]

I have tried to make integration because the velocity is changing
I said that T =$$\frac{x}{v}$$ >>
x = 9000 m
v = $$\int(313.5+0.607t)dt$$ and the limit of integration fro, -30 to 30
but it doesn't work

 

[hikaru1221]

I'm not sure what you mean by dv=vdt (deduced from your integral). Anyway, here is the way:
1 - We have dt = dx/v
2 - As v is given depending on T (note: t and T are different, one is time, the other is temperature), and we also have the variation law of T versus x (x is also height), we can derive v versus x.
Then, it's just simple math

 

[rwisz]

I would approach this problem by first understanding the concepts and equations involved. The given equation relates the speed of sound to temperature, which is affected by altitude. In order to solve this problem, we need to find the time interval required for the airplane sound to reach the ground, given the altitude and temperature conditions.

To start, we can use the given equation to calculate the speed of sound at the ground temperature of 30 degrees Celsius. Plugging in Tc=30, we get V=313.5 + 0.607(30) = 332.1 m/s.

Next, we need to determine the altitude at which the airplane is flying. We are given that the temperature decreases by 1 degree Celsius for every 150m rise in altitude. Since the ground temperature is 30 degrees Celsius, we can assume that the temperature at 9000m is 30-60= -30 degrees Celsius. This means that the airplane is flying at an altitude of 9000m.

Now, we can use the speed of sound at ground temperature and the altitude of the airplane to calculate the time interval. We know that speed = distance/time, so rearranging the equation, we get time = distance/speed. The distance in this case is the altitude of 9000m, and the speed is 332.1 m/s. Plugging these values into the equation, we get time = 9000m/332.1 m/s = 27.13 seconds.

Therefore, it would take approximately 27.13 seconds for the sound from the airplane flying at 9000m to reach the ground on a day when the ground temperature is 30 degrees Celsius. This time interval may vary depending on factors such as wind speed and direction, but based on the given information, this would be the estimated time.