Finding train speed from Doppler shift with 440Hz and 410Hz

[gigi9]

As a train APPROACHES a ringing crossing gate, Stacy, a passenger on the train, hears a frequency of 440 Hz from the bell. As the train RECEDES , she hears a frequency of 410 Hz. How fast is the train traveling? (formula: f = fo (v+ vo)/(v-vs))
f= frequency that she hears
fo= actual frequency
v= 330 m/s
vo= observer's frequency
vs= source's frequency
***Solve:
f1= 440 Hz (frequency that she hears as the train approaches the ringing gate)
f2= 410 Hz (frequency that she hears as the train approaches the ringing gate)
vs= 0 m/s
v= 330 m/s
vo=?
vo= [f1(v-vs)]/fo - v
As I got to this point, I was stuck b/c there was no fo, which is the actual frequency, so that I can plug into the equation. please show me how to do this problem...Thanks

 

[chroot]

By symmetry, f₀ is obviously halfway between 440 and 410 -- 425 Hz.

- Warren

 

[Alexander]

It actually is geometric average: f0=(f1xf2)^1/2=424.73 Hz, but here f1 anf f2 are so close that f0 is practically the same as arithmetic average 425 Hz anyway.

Gigi, use the Doppler equation for apparent frequency 2 times (one for approaching train and another for receeding), and you'll get TWO equations with 2 unknown variables (v, f0) - so you can solve for both.

To facilitate work, divide equations one by another and multiply them one by another (this way you'll immediately exclude one or the other unknown).