(a) 646.9 Hz
The formula for the Doppler effect is:
[tex]f'=\frac{v+v_o}{v-v_s}f[/tex]
where
f = 600 Hz is the real frequency of the sound
f' is the apparent frequency
v = 345 m/s is the speed of sound
[tex]v_o = 0[/tex] is the velocity of the observer (zero since it is stationary at the station)
[tex]v_s = +25 m/s[/tex] is the velocity of the source (the train), moving toward the observer
Substituting into the formula,
[tex]f'=\frac{345 m/s+0}{345 m/s-25 m/s}(600 Hz)=646.9 Hz[/tex]
(b) 20.1 m/s
In this case, we have
f = 600 Hz is the real frequency
f' = 567 Hz is the apparent frequency
Assuming the observer is still at rest,
[tex]v_o = 0[/tex]
so we can re-arrange the Doppler formula to find [tex]v_s[/tex], the new velocity of the train:
[tex]f'=\frac{v}{v-v_s}f\\\frac{f}{f'}=\frac{v-v_s}{v}\\\frac{f}{f'}v=v-v_s\\v_s = (1-\frac{f}{f'})v=(1-\frac{600 Hz}{567 Hz})(345 m/s)=-20.1 m/s[/tex]
and the negative sign means the train is moving away from the observer at the station.
