How long would it take a sound wave to travel 150m

[rpthomps]

Homework Statement

The sound speed in air at 0 degrees Celsius is 331 m/s, and for temperatures within a few tens of degrees of 0 degrees celsius it increases at the rate 0.590 m/s for every degree celsius increase in temperature. How long would it take a sound wave to travel 150 m over a path where the temperature rises linearly from 5 degrees to 15 degrees at the other end?

Homework Equations

d= 150 m


v=331+0.59T


$T=(\frac{1}{15})x+5\\dT=\frac{1}{15}dx$

The Attempt at a Solution

$dt=\frac { dx }{ v }\\T_{ total }=\int _{ 0 }^{ 150 }{ \frac { dx }{ v } }\\=\int _{ 0 }^{ 150 }{ \frac { 15dT }{ 331+0.59T } }\\=\frac { 15 }{ 0.59 } ln(331+0.59T){ | }_{ 0 }^{ 15 }=0.67$


Back of the book says .445 s

 

[SteamKing]

Homework Statement

The sound speed in air at 0 degrees Celsius is 331 m/s, and for temperatures within a few tens of degrees of 0 degrees celsius it increases at the rate 0.590 m/s for every degree celsius increase in temperature. How long would it take a sound wave to travel 150 m over a path where the temperature rises linearly from 5 degrees to 15 degrees at the other end?

Homework Equations

d= 150 m


v=331+0.59T


$T=(\frac{1}{15})x+5\\dT=\frac{1}{15}dx$

The Attempt at a Solution

$dt=\frac { dx }{ v }\\T_{ total }=\int _{ 0 }^{ 150 }{ \frac { dx }{ v } }\\=\int _{ 0 }^{ 150 }{ \frac { 15dT }{ 331+0.59T } }\\=\frac { 15 }{ 0.59 } ln(331+0.59T){ | }_{ 0 }^{ 15 }=0.67$


Back of the book says .445 s

Is the initial temperature 0° ?

 

[rpthomps]

Yup, that's it. the lower boundary should be 5 degrees. When I recalculate, I get the correct answer. Thank you sir!

 

[uknabm]

how thrilling is it that using a simple formula v = d/t, you can calculate how far a thunderstorm is from you! So exciting! I love Physics!