While blowing its horn of frequency 466 Hz, a car accelerates at 0.84 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

The car starts from rest by your side and moves away.

To determine how many seconds it takes for the frequency you hear to decrease by 5%, we need to consider two factors: the change in frequency due to the car moving away and the time it takes for sound to reach you.

First, let's calculate the change in frequency due to the car moving away. When an object is in motion, the frequency of the sound it produces changes relative to the observer's position.

The change in frequency can be determined using the Doppler effect formula:

Δf/f = v_r/c

Where:
Δf is the change in frequency,
f is the original frequency (466 Hz),
v_r is the relative velocity between the car and the observer,
c is the speed of sound in air (approximately 343 meters per second).

In this case, the car is moving away from you, so the relative velocity, v_r, is equal to the velocity of the car. Given that the car accelerates at 0.84 m/s² and starts from rest, we can use the equation of motion:

v = u + at

Where:
v is the final velocity,
u is the initial velocity,
a is the acceleration,
t is the time.

Since the initial velocity, u, is zero, we can rearrange the equation to solve for time:

t = v/a

Substituting the values, we get:

t = 0.84 m/s² / 0.84 m/s² = 1 second

So it takes 1 second for the car to reach a velocity of 0.84 m/s.

Now let's calculate the time it takes for sound to reach you. The speed of sound is approximately 343 meters per second, so the time it takes for sound to reach you is given by:

t_sound = distance / speed of sound

Since the car is moving away, the distance is equal to the relative velocity multiplied by the time it takes for the car to reach that velocity. Using the value of 0.84 m/s for the velocity and 1 second for the time, we get:

distance = 0.84 m/s * 1 s = 0.84 meters

Now, calculating the time it takes for sound to reach you:

t_sound = 0.84 meters / 343 m/s = 0.0025 seconds

Finally, we add the time it takes for sound to reach you to the time it takes for the car to reach a velocity of 0.84 m/s:

total time = 1 second + 0.0025 seconds = 1.0025 seconds

Therefore, it takes approximately 1.0025 seconds for the frequency you hear to decrease by 5%.