Calculating Speed of Approaching Police Car Using Beat Frequency

[mike91]

Homework Statement

Two police cars have identical sirens that produce a frequency of 660 Hz. A stationary listener is standing between two cars. One car is parked and the other is approaching the listener and both have their sirens on. The listener notices 8.7 beats per second. Find the speed of the approaching police car. (The speed of sound is 340 m/s.)

Homework Equations

f_beat = f_2 - f_1
v=340 m/s

The Attempt at a Solution

Ive been trying this one for a while.
Using the beat frequency (8.7 beats per second), I calculated f_2 (the frequency of the moving siren as heard by the listener) to be 668.7 Hz. I then used this in the equation for doppler shift, f_2=((v+v_L)/(v-v_s))f_1.
Plugging in v=340 m/s, v_L=0, f_2=668.7 Hz, f_1=660 Hz and solving for v_s, i get 3.9 m/s, which isn't right.
I have to be missing something small here, any help would be appreciated.

 

[Redbelly98]

The Attempt at a Solution

Ive been trying this one for a while.
Using the beat frequency (8.7 beats per second), I calculated f_2 (the frequency of the moving siren as heard by the listener) to be 668.7 Hz. I then used this in the equation for doppler shift, f_2=((v+v_L)/(v-v_s))f_1.
Plugging in v=340 m/s, v_L=0, f_2=668.7 Hz, f_1=660 Hz and solving for v_s, i get 3.9 m/s, which isn't right.
I have to be missing something small here, any help would be appreciated.

Welcome to Physics Forums.

Looks like your method is correct, and your answer is not that far off from the correct one. It's probably just an arithmetic or simple algebra mistake. If you post the details of your calculation, we could probably spot where the error is.

 

[mike91]

Thanks for the welcome, red.
Ok, so using the same methods listed above:
f_beat=f_2 - f_1
f_2 = 8.7 Hz + 660 Hz => f_2 = 668.7 Hz
For the doppler shift equation, the numerator (v+v_L) is just 340 m/s, since the listener is stationary. the denominator (v+v_s) is (340 - v_s), because the positive direction is from listener to source. f_1 = 660 Hz, so it all comes out to
668.7 = (660)*(340/(340-v_s)).
=> 1.01 = 340/(340-v_s)
=> 1.01(340-v_s) = 340
=> 340-v_s = 336.6
=> v_s = 3.4 m/s
Hm, if this looks ok Ill try this new answer. I must have rounded off an incorrect decimal place in my first attempts.

 

[Redbelly98]

Try expressing the "1.01" to, say, two more significant figures in your calculation.

 

[mike91]

So taking 668.7/660 and rounding off to 1.0132, I then get v_s = 4.43 m/s.

 

[Redbelly98]

Looks good. Probably 2 sig figs are justified in the final answer, given the original 8.7 Hz beat frequency.

 

[mike91]

That's it, thanks for the help!