In this equation, vd is the speed of the detector, and vs is the speed of the source. Also, fs is the frequency of the source, and fd is the frequency of the detector. If the detector moves toward the source, vd is positive. If the source moves toward the detector, vs is positive. A train moving away from a detector at 27 m/s blows a 290-Hz horn.
(a) What frequency is detected by a stationary train?
268.8378378
Hz
(b) What frequency is detected by a train moving away from the first train at a speed of 21 m/s?
To find the frequency detected by a train moving away from the first train at a speed of 21 m/s, we can use the formula for the Doppler effect:
fd = fs * (vd + vs) / (vd)
Given:
vd (speed of the detector) = 0 m/s (since the detector is stationary)
vs (speed of the source) = -21 m/s (since the source is moving away at a speed of 21 m/s)
fs (frequency of the source) = 290 Hz
Substituting the values into the formula, we have:
fd = 290 Hz * (0 m/s - 21 m/s) / (0 m/s)
fd = 290 Hz * (-21) / (0)
Here, we need to keep in mind that dividing by zero is undefined. In this case, since the detector is stationary, the frequency detected by the stationary train is not defined when the source is moving away at the same speed as the detector. Therefore, we cannot find the frequency detected by a stationary train if the source is moving away at a speed of 21 m/s.
To solve this problem, we can use the formula for the Doppler effect, which relates the observed frequency (fd) to the source frequency (fs) and the relative speeds of the source (vs) and the observer (vd).
(a) Since the first train is stationary, its speed (vs) is 0 m/s. The detector is also stationary, so its speed (vd) is 0 m/s as well. Therefore, we can use the formula:
fd = fs * (v + vd) / (v + vs)
Substituting fs = 290 Hz, vd = 0 m/s, and vs = 0 m/s, the equation simplifies to:
fd = fs
Thus, the frequency detected by a stationary train is the same as the frequency of the horn, which is 290 Hz.
(b) In this case, we have:
- fs = 290 Hz (source frequency)
- vd = 0 m/s (velocity of the detector)
- vs = -21 m/s (velocity of the source)
Using the same formula as before, we can calculate the detected frequency (fd):
fd = fs * (v + vd) / (v + vs)
Substituting the given values, we get:
fd = 290 * (343 + 0) / (343 + (-21))
= 290 * 343 / 322
Calculating this expression, we find that the frequency detected by a train moving away from the first train at a speed of 21 m/s is approximately 309.3164 Hz.
Hence, the frequency detected is approximately 309.3164 Hz.