Two golf carts have horns that emit sound with a frequency of 390 Hz. The golf carts are traveling toward one another. each traveling with a speed of 9.0 m/s with respect to the ground. What frequency do the drivers of the carts hear? The speed of sound at the golf course is 343 m/s.

Options

A. 390 HZ
B. 400 Hz
C. 410 Hz
D. 420 Hz

Answer

To solve the problem of finding the frequency heard by the drivers of the golf carts as they approach each other, we can use the Doppler effect formula. Here is a step-by-step solution:

### Step 1: Identify the parameters
- Frequency of the source (Fs) = 390 Hz
- Speed of sound in air (V) = 343 m/s
- Speed of each cart (Vo and Vs) = 9 m/s (both are moving toward each other)

### Step 2: Determine the Doppler effect formula
The formula for the frequency detected by the observer (Fo) when both the source and observer are moving towards each other is given by:

\[ F_o = \frac{V + V_o}{V - V_s} \times F_s \]

Where:
- \( F_o \) = frequency heard by the observer
- \( V \) = speed of sound in the medium
- \( V_o \) = speed of the observer (cart driver)
- \( V_s \) = speed of the source (horn of the other cart)
- \( F_s \) = frequency of the source

### Step 3: Substitute the values into the formula
Since both carts are moving towards each other, we will use the following values:
- \( V_o = 9 \, \text{m/s} \) (the observer's speed is positive)
- \( V_s = 9 \, \text{m/s} \) (the source's speed is negative)

Now substituting the values into the formula:

\[ F_o = \frac{343 + 9}{343 - 9} \times 390 \]

### Step 4: Calculate the numerator and denominator
Calculate the numerator and denominator separately:

Numerator:
\[ 343 + 9 = 352 \]

Denominator:
\[ 343 - 9 = 334 \]

### Step 5: Substitute back into the equation
Now substitute these values back into the equation:

\[ F_o = \frac{352}{334} \times 390 \]

### Step 6: Perform the division
Calculate the division:

\[ \frac{352}{334} \approx 1.053 \]

### Step 7: Multiply by the source frequency
Now multiply by the frequency of the source:

\[ F_o \approx 1.053 \times 390 \approx 411.07 \, \text{Hz} \]

### Step 8: Round to the nearest whole number
Rounding this value gives:

\[ F_o \approx 411 \, \text{Hz} \]

### Final Answer
The frequency heard by the drivers of the carts is approximately **411 Hz**.

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