# Dv/dt in a sound wave problem

1. Sep 11, 2010

### jwxie

1. The problem statement, all variables and given/known data

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?

2. Relevant equations

3. 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!