A child swinging on a swing set hears the sound of a whistle that is being blown directly in front
of her. At the bottom of her swing when she is moving toward the whistle, she hears the higher
pitch, and at the bottom of her swing, when she is moving away from the swing she hears a lower
pitch. The higher pitch has a frequency that is 5.0% higher than the lower pitch. What is the
speed of the child at the bottom of the swing?
I think I need to use the equation
freq observed=[(1-v child)/v sound]freq sound
I know the v sound =343 m/s but I have no idea even where to start on this question.
okay, so I set it up as a ratio:
[(1-v child)/v]*freq sound
OVER
[(1+v child)/v]*freq sound
ALL THAT =1.05
So I have 1.05=[(1-v child)/343]/[(1+v child)/343]
But I can't figure out how to manipulate this equation. I keep getting the v child as 341 m/s, and I know that kid is not moving that fast! How do I finish this equation?
I guess I gave up too soon!
To solve this problem, we can start by analyzing the Doppler effect formula:
freq observed = [(v sound + v observer) / (v sound + v source)] * freq source
Where:
- freq observed is the observed frequency by the observer
- freq source is the frequency of the source (whistle)
- v sound is the speed of sound (343 m/s)
- v observer is the velocity of the observer (child on the swing)
- v source is the velocity of the source (whistle)
We are given that the child hears a higher pitch (higher frequency) when moving towards the whistle, and a lower pitch (lower frequency) when moving away from it. Specifically, the higher pitch has a frequency that is 5.0% higher than the lower pitch.
Let's assign some variables to the velocities:
- v1 for the velocity of the child when moving towards the whistle
- v2 for the velocity of the child when moving away from the whistle
Therefore, the frequencies can be expressed as follows:
- freq observed1 = [(v sound + v1) / (v sound + 0)] * freq source
- freq observed2 = [(v sound + 0) / (v sound + v2)] * freq source
Given that the higher pitch is 5.0% higher than the lower pitch, we can write the equation as:
freq observed1 = 1.05 * freq observed2
Now, let's substitute freq observed1 and freq observed2 using the above equations:
[(v sound + v1) / v sound] * freq source = 1.05 * [(v sound + 0) / (v sound + v2)] * freq source
By canceling out freq source from both sides, we get:
(v sound + v1) / v sound = 1.05 * (v sound / (v sound + v2))
Simplifying further, we get:
v sound + v1 = 1.05 * (v sound + v2)
Now, let's isolate v1:
v1 = 1.05 * (v sound + v2) - v sound
Finally, we know that the child is at the bottom of the swing, where the speed is maximum. Therefore, the velocity of the child at the bottom of the swing is v2.
Hence, the speed of the child at the bottom of the swing is given by:
v2 = (v1 + v sound) / 1.05
By substituting the given value v sound = 343 m/s and rearranging the equation, we can find the speed of the child at the bottom of the swing.
To solve this problem, we can start by analyzing the situation. When the child is at the bottom of her swing, she hears a higher pitch as she swings towards the whistle and a lower pitch as she swings away from it.
The change in pitch, or frequency, is caused by the Doppler effect. The Doppler effect is the change in frequency of a wave (in this case, sound) due to the relative motion between the source of the wave (the whistle) and the observer (the child on the swing).
Let's assume the frequency of the whistle when the child is at the bottom of her swing and moving towards it is f1, and the frequency when she is at the bottom and moving away from it is f2. According to the problem, f1 is 5.0% higher than f2.
Now, we can use the Doppler effect equation to relate the observed frequency to the actual frequency of the sound. The equation you mentioned is correct:
freq observed = (1 - v(child) / v(sound)) * freq sound
Where:
- freq observed is the observed frequency by the child
- v(child) is the velocity of the child on the swing at the bottom
We know that the observed frequency when the child is moving towards the whistle (f1) is higher than the actual frequency (freq sound), and the observed frequency when the child is moving away from the whistle (f2) is lower than the actual frequency.
So, we can set up two equations:
f1 = (1 + v(child) / v(sound)) * freq sound (1)
f2 = (1 - v(child) / v(sound)) * freq sound (2)
We want to find the speed of the child at the bottom of the swing, which is v(child) in the equations. To solve for v(child), we need to eliminate freq sound from the equations.
Since the problem provides the value for the speed of sound in air (v(sound) = 343 m/s), we can substitute it into equations (1) and (2).
f1 = (1 + v(child) / 343) * freq sound (1)
f2 = (1 - v(child) / 343) * freq sound (2)
Next, we can use the information that the higher pitch is 5.0% higher than the lower pitch. Mathematically, this can be stated as:
f1 = f2 * (1 + 0.05)
Now, we can substitute this relationship into the equations:
f2 * (1 + 0.05) = (1 + v(child) / 343) * freq sound (1)
f2 = (1 - v(child) / 343) * freq sound (2)
Now we have a system of two equations with two unknowns: f2 and v(child).
Simplifying equation (1), we get:
1 + 0.05 = 1 + v(child) / 343
0.05 = v(child) / 343
Solving for v(child), we find:
v(child) = 343 * 0.05
v(child) = 17.15 m/s
Therefore, the speed of the child at the bottom of the swing is 17.15 m/s.