Dv/dt in a sound wave problem

[jwxie]

Homework Statement

The speed of sound in air (in meters per second) depends on temperature according to the approximate expression
v = 331.5 + 0.607T

where T is the Celsius temperature. In dry air, the temperature decreases about 1C for every 150m rise in altitude.

a. Assume this change is constant up to an altitude of 9000m. What time interval is required for the sound from an airplane flying at 9000m to reach the ground on a day when the ground temperature is 30C?

Homework Equations

The Attempt at a Solution

First I found the changes in T from 9000m to ground level.
9000m / 150m = 60, which means at 9000m we have -30C.

Then I was stuck because I didn't know how to take the integration.
Initially I thought about dV/dt because we are taking about changes in time.

But the formula v = 331.5 + 0.607T only provided T the temperature. If I coerce to take the integral, I would end up the unknown time (t) in the formula, which is exactly what I need to find out.

My professor showed us to derive the integral in the form

$$\[\frac{\mathrm{d} v}{\mathrm{d} t} = \frac{\mathrm{d} v}{\mathrm{d} T} \frac{\mathrm{d} T}{\mathrm{d} x} \frac{\mathrm{d} x}{\mathrm{d} t}\]$$

I was shocked! Right. The changes in t in velocity depends on the changes in temperature and changes in altitude.

So my first question is:
(1) What would you do when you first read this problem? How would you interpret the dv/dt when you try to derive a valid integrand yourself?
What I am interested in is how do people actually come up with this? I hope I didn't confuse you.


(2) An alternative is to take the average of the velocity. Since the changes is constant, so we can use the short cut 1/2 (changes in V) = V_average

But why do we have to take average?


(3) Assuming the changes is not constant: maybe at 6000m the change in T is 1C per 200m thereafter. How would this affect the second method?

(4) If changes is not constant throughout as in #3, when we use integration, we have to break it into 2 integrals, right?


What I really want to understand is how to become a better problem solver. I am not so smart IMO.
Thank you!

 

[anathema]

Dear forum post author,

Thank you for your question. I would approach this problem by first understanding the given equation for the speed of sound in air and its dependence on temperature. I would then use this equation to calculate the speed of sound at different altitudes, using the given information that the temperature decreases by 1C for every 150m rise in altitude.

To solve for the time interval required for the sound from an airplane flying at 9000m to reach the ground, I would use the formula for distance traveled (distance = speed x time) and solve for time. In this case, the distance traveled is 9000m and the speed of sound is given by the equation v = 331.5 + 0.607T, where T is the temperature at the given altitude. The temperature at 9000m can be calculated using the information that the temperature decreases by 1C for every 150m rise in altitude.

Regarding your questions:

1) When I first read this problem, I would interpret dv/dt as the rate of change of velocity with respect to time. In order to derive a valid integrand, I would use the formula for distance traveled (distance = speed x time) and solve for time, as mentioned above.

2) Taking the average of velocity is a shortcut method that can be used when the changes are constant. This is because the average velocity is equal to the final velocity divided by 2, when the initial velocity is 0. However, in this problem, the changes in temperature and altitude are not constant, so we cannot use this shortcut method.

3) If the changes are not constant, we cannot use the shortcut method of taking the average of velocity. Instead, we would have to use the formula for distance traveled (distance = speed x time) and solve for time at each altitude.

4) Yes, if the changes are not constant throughout, we would have to break the integral into smaller intervals and solve for time at each interval.

To become a better problem solver, I would suggest practicing solving different types of problems and understanding the underlying concepts and equations. It takes time and practice to become a good problem solver, so don't be too hard on yourself. Keep practicing and seeking help when needed, and you will improve over time.

I hope this helps. Best of luck with your studies!