1. A whistle you use to call your hunting dog has a frequency of 21 kHz, but your dog is ignoring it. You suspect the whistle may not be working, but you can't hear sounds above 20 kHz. To test it, you ask a friend to blow the whistle and hop on their moped. In which direction should they ride (toward or away from you) to know if the whistle is working? Explain in words or with a diagram.
2.The railroad crossing lights turn red, so McKayla and her sister must stop and wait for the train to pass by. As they wait, McKayla's sister Kylie grabs her phone and uses an app to measure the frequency of the approaching train's horn. The app reads 429Hz. Assuming the train's original horn frequency is 400Hz and the speed of sound is 330m/s, how fast is the train going in m/s and miles per hour?
3. Going back to the dog whistle in question 1, what is the minimum riding speed needed to be able to hear the whistle? Remember, you can assume the following things: The whistle you use to call your hunting dog has a frequency of 21.0 kHz, but your dog is ignoring it. You suspect the whistle may not be working, but you can't hear sounds above 20.0 kHz. The speed of sound is 330m/s at the current air temperature.
1.
The apparent frequecy should be 20kHz i.e.lower than the actual freq. of 21kHz. Therefore the sound source should move away from you.
If the rider speed is U -
F' = v*F/(v+U)
20*10^3= 330*21*10^3/(330+U)
330+U= 330*21/20
U = 330*[21/20 -1]= 330*1/20
= 16.5m/s (min.riding speed)
2.
Train is approaching the observer.
429=330*400/[330-U]
330-U=330*400/429
U = 330[1-400/429]=22.3m/s (train's speed)
1. To test if the whistle is working, your friend should ride their moped away from you. If the whistle is working, the frequency of the sound waves will decrease as they travel towards you due to the Doppler effect. If the whistle is not working, the frequency will remain the same regardless of the direction your friend is riding.
2. To find the speed of the train, we can use the formula for the Doppler effect:
f' = (v + vr) / (v - vs) * f
Where f' is the measured frequency, v is the speed of sound, vr is the velocity of the receiver (Kylie and McKayla) towards the source (train), vs is the velocity of the source (train) towards the receiver, and f is the original frequency.
In this case, f' = 429 Hz, v = 330 m/s, vr = 0 (assuming Kylie and McKayla are not moving towards the train), vs is what we need to find, and f = 400 Hz.
Plugging in the values and solving for vs:
429 = (330 + 0) / (330 - vs) * 400
429(330 - vs) = 330 * 400
141570 - 429vs = 132000
429vs = 9570
vs = 9570 / 429
vs ≈ 22.3 m/s
To convert the speed to miles per hour, we can use the conversion factor: 1 m/s = 2.237 mph.
So, the train is traveling at approximately 22.3 m/s or 49.9 mph.
3. To find the minimum riding speed needed to hear the whistle, we can use the formula for the Doppler effect:
f' = (v + vr) / (v - vs) * f
In this case, f' is the frequency of the whistle (21 kHz), v is the speed of sound (330 m/s), vr is the velocity of the receiver (you) towards the source (whistle), vs is what we need to find, and f is the original frequency (20 kHz).
Plugging in the values and solving for vs:
21,000 = (330 + vr) / (330 - vs) * 20,000
21,000 * (330 - vs) = 20,000 * (330 + vr)
6,930,000 - 21,000vs = 6,600,000 + 20,000vr
-21,000vs = -330,000 + 20,000vr
vs = (330,000 - 20,000vr) / 21,000
To hear the whistle, the frequency observed needs to be equal to or greater than 20 kHz. Since you can't hear sounds above 20 kHz, the observed frequency (f') needs to be equal to or greater than 20 kHz.
So, the minimum riding speed needed to be able to hear the whistle depends on the velocity of the receiver (you) towards the source (whistle), which is represented as vr. If vr is greater than or equal to 330, then vs will be less than or equal to 0 (to achieve a frequency of 20 kHz or higher). If vr is less than 330, then the velocity of the source (whistle) needs to be greater than 0 for vs to be less than 0 and achieve a frequency of 20 kHz or higher.
1. To test if the whistle is working, your friend should ride their moped away from you. This is because as they ride away, the sound waves produced by the whistle will be stretched and their frequency will decrease. If the whistle is working, the frequency of the sound wave will decrease to a level where you can hear it, even though it is below 20 kHz. If the whistle is not working, the frequency will remain above 20 kHz and you still won't be able to hear it.
2. To find the speed of the train, we can use the formula:
Speed = (Observed frequency - Original frequency) / (Original frequency) * Speed of Sound
Speed = (429 Hz - 400 Hz) / (400 Hz) * 330 m/s
Speed = (29 Hz) / (400 Hz) * 330 m/s
Speed = 330 * (29/400) m/s
Speed = 24.075 m/s
To convert the speed to miles per hour, we can use the conversion factor:
1 mile = 1609.34 meters,
1 hour = 3600 seconds.
Speed = (24.075 m/s) * (3600 s/1 hr) * (1 mile/1609.34 m)
Speed = 53.9 mph (approximately)
So, the train is going approximately 24.075 m/s or 53.9 mph.
3. To determine the minimum riding speed needed to hear the whistle, we can use the Doppler effect equation:
f' = fo * (v + vr) / (v + vs)
where:
fo = original frequency of the whistle
v = speed of sound
vr = speed of the rider
vs = speed of the source (whistle)
Given:
fo = 21.0 kHz = 21,000 Hz
v = 330 m/s
We want to find the minimum riding speed (vr) needed to hear the whistle, so we set f' to 20.0 kHz:
20,000 Hz = 21,000 Hz * (330 m/s + vr) / (330 m/s + 0)
Simplifying the equation:
20,000 Hz = 21,000 Hz * (330 m/s + vr) / (330 m/s)
Cross-multiplying:
20,000 Hz * 330 m/s = 21,000 Hz * (330 m/s + vr)
6,600,000 Hz * m/s = 6,930,000 Hz * m/s + 21,000 Hz * vr
Subtracting 6,930,000 Hz * m/s from both sides:
(6,600,000 Hz * m/s) - (6,930,000 Hz * m/s) = 21,000 Hz * vr
-330,000 Hz * m/s = 21,000 Hz * vr
Dividing both sides by 21,000 Hz:
-330,000 Hz / 21,000 Hz = vr
-15.71 m/s = vr
So, the minimum riding speed needed to hear the whistle is approximately 15.71 m/s, which is equivalent to about 35.17 mph.
1. To determine if the whistle is working, your friend should ride away from you. As your friend moves away from you, the sound waves emitted by the whistle will undergo a phenomenon called the Doppler effect. The Doppler effect causes a change in the frequency of a wave (in this case, sound) perceived by an observer when there is relative motion between the source of the wave and the observer. When the source of the wave moves away from the observer, the observed frequency decreases.
In this situation, since you cannot hear sounds above 20 kHz, you won't be able to hear the whistle if it is emitting sound with a frequency of 21 kHz. However, if your friend rides away from you and you can suddenly hear the whistle, it means the frequency of the whistle has decreased due to the Doppler effect, indicating that the whistle is indeed working.
2. To calculate the speed of the train, we can use the Doppler effect equation:
f' = (v + vr) / (v + vs) * f
where f' is the observed frequency, v is the speed of sound, vr is the velocity of the receiver, vs is the velocity of the source, and f is the original frequency.
In this case, the observed frequency (f') is 429 Hz, the original frequency (f) is 400 Hz, and the speed of sound (v) is 330 m/s.
To calculate the velocity of the receiver (vr), we can assume McKayla and her sister are stationary, so vr would be 0.
Now we can rearrange the equation to solve for the velocity of the source (vs), which is the velocity of the train:
vs = (f' / f - 1) * v
Plugging in the values, we get:
vs = (429 / 400 - 1) * 330
Calculating this, we find that vs is approximately 35.85 m/s.
To convert this speed to miles per hour, we know that 1 mile is equal to 1609.34 meters and 1 hour is equal to 3600 seconds. Therefore:
speed in miles per hour = (35.85 * 3600) / 1609.34
Calculating this, we find that the speed of the train is approximately 80.17 mph.
3. To determine the minimum riding speed needed to hear the dog whistle, we need to consider the Doppler effect again. Since you cannot hear sounds above 20 kHz and the whistle has a frequency of 21.0 kHz, the frequency needs to be decreased enough for you to hear it.
Using the same Doppler effect equation as before:
f' = (v + vr) / (v + vs) * f
where f' is the observed frequency, v is the speed of sound, vr is the velocity of the receiver (which is 0 in this case since you are stationary), vs is the velocity of the source (riding speed of your friend), and f is the original frequency (21.0 kHz).
We want to find the minimum riding speed (vs) that will make the observed frequency (f') equal to or slightly below 20.0 kHz.
We can rearrange the equation to solve for the velocity of the source:
vs = (f' / f - 1) * v
Plugging in the values, we have:
vs = (20.0 / 21.0 - 1) * 330
Calculating this, we find that vs is approximately -14.29 m/s.
Since velocity cannot be negative in this context, it means that your friend would have to ride at a speed of at least 14.29 m/s (or approximately 31.99 mph) towards you for you to be able to hear the whistle.