The observed frequency of a train whistle is 513Hz as it approaches a stationary observer and is 432Hz as it recedes.
Find the speed of the train
Approaching
fo = fs[1/(1-vs/v)]
Receding
fo = fs[1/(1+vs/v)]
set them equal and solve for vs
To find the speed of the train, we will use the formula for the Doppler effect. The Doppler effect describes the change in frequency of a wave (in this case, sound) due to the relative motion between the source of the sound (the train) and the observer. The formula for the Doppler effect is:
Δf/f = v/c
Where:
Δf is the change in frequency
f is the original frequency
v is the velocity of the source (in this case, the train)
c is the speed of sound in the medium (in this case, assume it to be 343 m/s, which is the speed of sound in air at 20°C)
In this case, we have two different frequencies, one as the train approaches (f_a) and one as it recedes (f_r). Using these frequencies, we can set up two equations and solve for the velocity (v) of the train.
For the approaching frequency:
Δf_a = f_a - f
Δf_a = 513 Hz - f
For the receding frequency:
Δf_r = f - f_r
Δf_r = f - 432 Hz
Setting these equations to be equal to our Δf/f ratio (since they represent the same velocity), we have:
Δf_a / f = v / c
Δf_r / f = -v / c (negative sign because the velocity is in the opposite direction)
Rearranging these equations to solve for v, we get:
v = c * (Δf_a / f)
v = -c * (Δf_r / f)
Since we have two equations, we can take the average of the two velocities to find the actual velocity of the train:
v = (c * (Δf_a / f) - c * (Δf_r / f)) / 2
Now we can substitute the given values and solve for v:
c = 343 m/s
Δf_a = 513 Hz - 432 Hz = 81 Hz
Δf_r = 432 Hz - 513 Hz = -81 Hz
Substituting these values into the equation, we get:
v = (343 * (81 / f) - 343 * (-81 / f)) / 2
Now, we need to find the original frequency (f). We can use either the approaching or receding frequency, so let's choose the approaching frequency:
Δf_a = f_a - f
81 Hz = 513 Hz - f
f = 513 Hz - 81 Hz
f = 432 Hz
Now we can substitute this value back into the equation to find v:
v = (343 * (81 / 432 Hz) - 343 * (-81 / 432 Hz)) / 2
v = (343 * 0.1875 - 343 * -0.1875) / 2
v = (64 + 64) / 2
v = 128 / 2
v = 64 m/s
Therefore, the speed of the train is 64 m/s.