How Does the Doppler Effect Help Determine the Speed of Sound?

[calculator20]

Homework Statement

Problem: a car is moving away from a stationary observer at 25ms^-1. It emits a frequency of 810Hz and the observer hears an apparent frequency of 762Hz. What is the speed of sound in air at that time?

Homework Equations

The equation for a source moving away is

f' = f(v/(v+us)) where us is speed of source.

The Attempt at a Solution

There are 2 v's and I need just one, I have no idea where to start, please can you help?

 

[Doc Al]

There are 2 v's and I need just one, I have no idea where to start, please can you help?

Of the two speeds, one is given. Just solve for the other.

 

[calculator20]

Yes I know I have us but there are two v's in the equation and I don't know how to rearrange it to get one v on its own?

 

[Doc Al]

Yes I know I have us but there are two v's in the equation and I don't know how to rearrange it to get one v on its own?

Ah, so it's the algebra that's messing you up. First thing to do is get rid of that denominator on the right hand side. How can you do that?

 

[calculator20]

Multiply by it giving this?

f'v+fus = fv

 

[Doc Al]

Multiply by it giving this?

f'v+fus = fv

Almost. That second term on the left should have an f', not an f.

 

[calculator20]

Oh yes, sorry.

f'v + f'us = fv

I still can't see how I set it down to just one v without somehow cancelling both out along the way?

 

[Doc Al]

Oh yes, sorry.

f'v + f'us = fv

I still can't see how I set it down to just one v without somehow cancelling both out along the way?

Keep the term with us where it is, but move all the others to the right. Then you can solve for us.

 

[calculator20]

But I'm solving for v?

 

[Doc Al]

But I'm solving for v?

No, v is the speed of the car. You need to solve for u_s, which is the speed of sound.

 

[calculator20]

It's definitely v I need to find. According to my textbook us is the moving source velocity. In the other equation it uses uo as moving observer velocity.

 

[Doc Al]

It's definitely v I need to find. According to my textbook us is the moving source velocity. In the other equation it uses uo as moving observer velocity.

You're right! (I must be losing my mind. Sorry about that!)

So just collect all the terms with v in them to one side. Then you can isolate v. [Using ax + bx = (a+b)x]

 

[calculator20]

f'us=fv-f'v

f'us =v(f-f')

v=f'us/(f-f')

Is that right?

 

[Doc Al]

f'us=fv-f'v

f'us =v(f-f')

v=f'us/(f-f')

Is that right?

Perfect!

 

[calculator20]

Excellent! Thank you so much for your help! :)