.Two automobiles are equipped with the same single-frequency horn. When one is at rest and the other is moving toward an observer at 14 m/s, a beat frequency of 5.1 Hz is heard. What is the frequency the horns emit? Assume T = 20°C.

The frequency difference is 5.1 Hz

(1 + V/a)*f - f = (V/a)*f = 5.1
a is the sound speed (340 m/s), and
V = 14 m/s

Solve for f.

f = 124 Hz

I did not use the exact formula for a doppler-shifted moving source, but this should be close enough.

To solve this problem, we can use the formula for the beat frequency:

Beat Frequency (f_beat) = |f1 - f2|

Where f1 and f2 are the frequencies of the two sources.

We are given:
- Beat frequency (f_beat) = 5.1 Hz
- Speed of the moving automobile (v) = 14 m/s

To find the frequency the horns emit, we need to find the individual frequencies of the two sources. Let's assume f1 is the frequency of the stationary automobile, and f2 is the frequency of the moving automobile.

The formula to calculate the observed frequency when a source is moving is given by the Doppler effect equation:

f_observed = (v_sound + v_observer) / (v_sound + v_source) * f_emitted

Where:
- v_sound is the speed of sound
- v_observer is the speed of the observer
- v_source is the speed of the source (automobile)
- f_emitted is the frequency emitted by the source

We are given:
- v_observer = 0 m/s (since the observer is at rest)
- v_sound = approximately 343 m/s at 20°C
- v_source = -14 m/s (negative sign signifies that it is moving towards the observer)

Substituting these values into the Doppler effect equation, we get:

f_observed = (343 + 0) / (343 - 14) * f_emitted
f_observed = 343 / 329 * f_emitted
f_observed = 1.043 * f_emitted

Now, let's rearrange the equation to solve for f_emitted:

f_emitted = f_observed / 1.043

Since the beat frequency is the difference between the observed frequency and the frequency emitted by the stationary source, we can write:

f_beat = f_observed - f_emitted
f_beat = f_observed - (f_observed / 1.043)

Now, substitute the given value of f_beat and solve for f_observed:

5.1 = f_observed - (f_observed / 1.043)
5.1 = f_observed * (1 - 1/1.043)

To isolate f_observed, divide both sides of the equation by (1 - 1/1.043):

f_observed = 5.1 / (1 - 1/1.043)

Calculate the value of f_observed:

f_observed = 5.1 / (1 - 1/1.043) ≈ 5.2359 Hz

Finally, substitute the value of f_observed into the equation for f_emitted to find the frequency emitted by the horns:

f_emitted = f_observed / 1.043
f_emitted = 5.2359 / 1.043 ≈ 5.0149 Hz

Therefore, the frequency that the horns emit is approximately 5.0149 Hz.

To find the frequency the horns emit, we need to understand the concept of beat frequency and the Doppler effect.

The beat frequency is the difference between the frequencies of two sound waves that interfere with each other. In this case, one horn is at rest, so its frequency is constant. The other horn is moving towards the observer, resulting in a change in frequency due to the Doppler effect.

The Doppler effect describes the change in frequency of a wave as perceived by an observer moving relative to the source of the wave. When the source is moving towards the observer, the frequency increases.

Let's denote the frequency of the horn at rest as f_0, and the frequency of the horn in motion as f.

The formula for the beat frequency (f_beat) is given by:
f_beat = |f - f_0|

In this case, the beat frequency is given as 5.1 Hz.

We know that the speed of sound in air at 20°C is approximately 343 m/s.

The Doppler effect formula for sound moving towards an observer is:
f = (v + vo)/(v + vs) * f_0

Where:
f = frequency perceived by the observer
v = speed of sound
vo = velocity of the observer (in this case, zero since the observer is at rest)
vs = velocity of the source (in this case, 14 m/s since the other automobile is moving towards the observer)

Rearranging the formula, we get:
f_0 = (v + vs)/(v + vo) * f

Substituting the given values:
f_0 = (343 + 14)/(343 + 0) * f
f_0 = 357/343 * f

Using the equation for beat frequency, we know that:
f_beat = |f - f_0|

Substituting the values:
5.1 = |f - (357/343 * f)|

Now, we can solve this equation to find the value of f.

|f - (357/343 * f)| = 5.1

This equation is a bit complicated to solve analytically, but we can use numerical methods to find an approximate solution. One way is to use a graphing calculator or software.

Alternatively, we can use iterative methods to find the root of the equation through trial and error. Starting with an initial guess for f, we can substitute it into the equation, calculate the left-hand side, and adjust the guess until we get a value close to the right-hand side (5.1 Hz).

By repeating this process, we can find the value of f that satisfies the equation.