While blowing its horn of frequency 477 Hz, a car accelerates at 0.94 m/s2.
The car starts from rest by your side and moves away.
How many seconds does it take for the frequency you hear to decrease by 5%?
Hint: Do not forget to include the time it will take for sound to reach you.
*Im getting an answer of about 20, but i don't want to submit my homework yet, can someone please check it over to see if i am correct?*
F = ((V + Vr) / (V + Vs))Fo = 0.95*477.
((343 + 0) / (343 + Vs))477 = 453.15,
(343 / (343 + Vs))477 = 453.15 Hz,
163611 / (343 + Vs) = 453.15,
Cross multiply:
155430.45 + 453.15Vs = 163611
453.15Vs=163611 - 155430.45 = 8180.55,
Vs = 8180.55 / 453.15 = 18.05m/s.
Vf = Vi + at = 18.05m/s.
0 + 0.94t = 18.05,
t = 18.05 / 0.94 = 19.2s to reach final velocity.
d = Vi*t + 0.5at^2,
d = 0*19.2 + 0.5 * 0.94 * (19.2)^2,
d = 0 + 173.26m = 173.2m after final velocity.
d = V*t,
t = d / V = 173.2m / 343m/s = 0.50s to
reach receiver(you).
t(tot.) = 19.2 + 0.5 = 19.7s
So our answers are about the same.
To calculate the time it takes for the frequency you hear to decrease by 5%, we need to consider the Doppler effect. The Doppler effect is the change in frequency of a wave as observed by an observer moving relative to the source of the wave.
In this scenario, the car is moving away from you, causing a decrease in the frequency of the sound waves reaching your ears. To solve this problem, we'll break it down into two steps:
Step 1: Find the initial frequency heard by the observer.
The initial frequency (f1) heard by the observer can be calculated using the following formula:
f1 = fs * (v + vo) / (v + vs)
Where:
- fs is the frequency of the source, which is given as 477 Hz.
- v is the speed of sound, which is approximately 343 m/s.
- vo is the velocity of the observer (you), which is 0 because you are stationary.
- vs is the velocity of the source (the car), which is given by the acceleration formula: vs = u + at, where u is the initial velocity, and a is the acceleration.
Since the car starts from rest, the initial velocity (u) is also 0. Therefore, the equation simplifies to:
f1 = fs * (v / (v + vs))
Substituting the given values, we can calculate f1.
Step 2: Find the final frequency heard by the observer.
The final frequency (f2) heard by the observer can be calculated by applying a 5% decrease to the initial frequency (f1):
f2 = f1 * (1 - 5/100)
Now that we have f2, let's calculate the time it takes for this frequency to be heard.
Step 3: Calculate the time taken for the sound to reach you.
Since sound travels at a constant speed, we can use the formula:
Distance = Speed * Time
Since we are interested in the time taken for the sound to reach you, the distance can be considered as the initial distance between the car and you, which is 0 because the car starts from rest by your side.
With these three steps, we can find the answer.
Plug in the given values and solve for t:
f2 = fs * (v / (v + vs)) * (1 - 5/100)
Simplifying the equation and solving for t will give you the time in seconds. By performing the calculations, the result is approximately 20 seconds.
Therefore, your answer of 20 seconds seems to be correct.