Alice holds a small battery operated device used for tuning instruments that emits the frequency of middle C (262 Hz) while walking with a constant speed of 3.13 m/s toward a building which presents a hard smooth surface and hence reflects sound well. (Use 343 m/s as the speed of sound in air.)
(a) Determine the beat frequency Alice observes between the device and its echo. (Enter your answer to at least 1 decimal place.)
(b) Determine how fast Alice must walk away from the building in order to observe a beat frequency of 6.59 Hz.
(a) Well, it sounds like Alice is really playing her device in sync with the universe! To determine the beat frequency, we need to find the difference between the frequency of the device and the frequency of its echo.
The frequency of the device is 262 Hz, and since it is approaching the building, the apparent frequency will be higher. This change in frequency is given by the Doppler effect formula:
f' = f * ((v + v0) / (v - vs)),
where f' is the apparent frequency, f is the actual frequency, v is the speed of sound in air, v0 is the speed of Alice, and vs is the speed of the building (which we'll assume is 0 since it's not moving).
Plugging in the values, we have:
f' = 262 * ((343 + 3.13) / (343 - 0)).
Calculating this, we get:
f' = 267.348 Hz.
Now, let's calculate the beat frequency by subtracting the actual frequency from the apparent frequency:
Beat frequency = f' - f = 267.348 - 262 = 5.348 Hz.
So, the beat frequency that Alice observes is approximately 5.3 Hz.
(b) Oh, Alice wants to spice up her musical walks, I see! To determine the speed Alice needs to walk away from the building, we'll use a similar approach.
We can use the same Doppler effect formula as before, but now we'll solve for v0, the speed of Alice:
v0 = ((f' / f) - 1) * v.
Plugging in the given values, we have:
v0 = ((6.59 / 262) - 1) * 343.
Calculating this, we get:
v0 = -0.908 m/s.
Wait, negative speed? That means Alice needs to walk backwards to achieve a beat frequency of 6.59 Hz! Maybe she should consider becoming a moonwalker instead.
But if we take the magnitude of the speed, Alice needs to walk at a speed of approximately 0.908 m/s away from the building to observe a beat frequency of 6.59 Hz.
Now, let's hope she doesn't stumble and trip on her way. Stay in tune, Alice!
To determine the beat frequency Alice observes between the device and its echo, we need to first calculate the frequency of the echo. The frequency of the echo is affected by the Doppler effect, which accounts for the motion of the source (Alice) and the observer (Alice again).
The observed frequency, \( f' \), of a moving source is given by:
\[ f' = \frac{f \cdot (v + v_s)}{v} \]
where:
- \( f \) is the frequency of the source (262 Hz in this case)
- \( v \) is the speed of sound in air (343 m/s)
- \( v_s \) is the speed of the source (3.13 m/s)
Substituting the given values into the equation, we can calculate the observed frequency:
\[ f' = \frac{262 \times (343 + 3.13)}{343} \]
Simplifying the equation gives:
\[ f' \approx 264.375 \, \text{Hz} \]
To calculate the beat frequency, we subtract the frequency of the echo from the frequency of the device:
\[ \text{Beat Frequency} = f - f' \]
Substituting the values into the equation:
\[ \text{Beat Frequency} = 262 \, \text{Hz} - 264.375 \, \text{Hz} \]
Simplifying:
\[ \text{Beat Frequency} \approx -2.375 \, \text{Hz} \]
Since the beat frequency is negative, it means that Alice observes a beat frequency of 2.375 Hz when walking toward the building.
Now let's move on to part (b) - determining how fast Alice must walk away from the building to observe a beat frequency of 6.59 Hz.
Using the same formula as before, but this time solving for the source speed (\( v_s \)), we have:
\[ v_s = v \left( \frac{f - f'}{f} \right) \]
Substituting the given values:
\[ v_s = 343 \left( \frac{262 - 6.59}{262} \right) \]
Simplifying:
\[ v_s \approx 332.906 \, \text{m/s} \]
Therefore, Alice must walk away from the building with a speed of approximately 332.906 m/s to observe a beat frequency of 6.59 Hz.
To determine the beat frequency Alice observes between the device and its echo, we need to consider the Doppler effect. The Doppler effect occurs when there is relative motion between the source of the sound and the observer.
The formula for the observed frequency due to the Doppler effect is given by:
f_obs = f_source * (v_sound + v_observer) / (v_sound + v_source)
Where:
f_obs is the observed frequency,
f_source is the frequency of the source,
v_sound is the velocity of sound in air,
v_observer is the velocity of the observer, and
v_source is the velocity of the source.
In this case, Alice is the observer, and she is moving towards the building. The device emits the frequency of middle C, which is 262 Hz. The speed of sound in air is 343 m/s.
(a) To find the beat frequency Alice observes between the device and its echo, we need to consider the reflection of sound from the building. When the sound is reflected, the frequency remains the same, but its sign changes. Therefore, the frequency observed would be the same as the frequency emitted, but its sign would change.
Using the formula, we can calculate the observed frequency:
f_obs = -f_source * (v_sound + v_observer) / (v_sound + v_source)
Substituting the given values:
f_obs = -262 Hz * (343 m/s + 3.13 m/s) / (343 m/s - 3.13 m/s)
Simplifying the expression:
f_obs = -262 Hz * 346.13 m/s / 339.87 m/s
f_obs = -267.13 Hz
Therefore, Alice observes a beat frequency of -267.1 Hz (with a negative sign indicating a decrease in frequency) between the device and its echo.
(b) In order to observe a beat frequency of 6.59 Hz, we need to find the velocity of Alice walking away from the building.
Using the same formula as before:
f_obs = -f_source * (v_sound + v_observer) / (v_sound + v_source)
Substituting the given values:
6.59 Hz = -262 Hz * (343 m/s - 3.13 m/s) / (343 m/s + v_source)
Simplifying the expression:
6.59 Hz * (343 m/s + v_source) = -262 Hz * (343 m/s - 3.13 m/s)
(343 m/s + v_source) / (343 m/s - 3.13 m/s) = -262 Hz / 6.59 Hz
(343 m/s + v_source) / 340.87 m/s = -39.7
343 m/s + v_source = -39.7 * 340.87 m/s
v_source = -39.7 * 340.87 m/s - 343 m/s
v_source = -13496.079 m/s
Therefore, Alice must walk away from the building at a velocity of approximately 13496.1 m/s in order to observe a beat frequency of 6.59 Hz.