An aircraft carrier has a speed of 13.0 m/s relative to the water. A jet is catapulted from the deck and has a speed of 67.0 m/s relative to the water. The engines produce a 1550-Hz whine, and the speed of sound is 343 m/s. What is the frequency of the sound heard by the crew on the ship?

Options

A. 1200 Hz
B. 1450 Hz
C. 1180 Hz
D. 1350 Hz

Answer

To solve the problem, we need to use the Doppler effect formula to find the frequency of sound heard by the crew on the aircraft carrier. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the given values
- Speed of sound in air, \( v = 343 \, \text{m/s} \)
- Speed of the aircraft carrier (observer), \( v_o = 13.0 \, \text{m/s} \)
- Speed of the jet (source), \( v_s = 67.0 \, \text{m/s} \)
- Frequency of the sound produced by the jet (source frequency), \( f_s = 1550 \, \text{Hz} \)

### Step 2: Determine the direction of motion
The aircraft carrier is moving at 13.0 m/s towards the direction of the jet, while the jet is moving away from the carrier at 67.0 m/s.

### Step 3: Apply the Doppler effect formula
The formula for the frequency detected by the observer is given by:

\[ f_o = f_s \times \frac{v + v_o}{v - v_s} \]

Where:
- \( f_o \) = frequency heard by the observer (crew on the carrier)
- \( f_s \) = frequency of the source (jet)
- \( v \) = speed of sound in the medium (air)
- \( v_o \) = speed of the observer (aircraft carrier)
- \( v_s \) = speed of the source (jet)

### Step 4: Substitute the values into the formula
Substituting the known values into the formula:

\[ f_o = 1550 \times \frac{343 + 13}{343 - 67} \]

### Step 5: Calculate the numerator and denominator
Calculating the numerator:

\[ 343 + 13 = 356 \, \text{m/s} \]

Calculating the denominator:

\[ 343 - 67 = 276 \, \text{m/s} \]

### Step 6: Plug the values back into the equation
Now substituting back into the equation:

\[ f_o = 1550 \times \frac{356}{276} \]

### Step 7: Calculate the frequency
Now, calculate the fraction:

\[ \frac{356}{276} \approx 1.287 \]

Now, multiply by the source frequency:

\[ f_o \approx 1550 \times 1.287 \approx 1995.85 \, \text{Hz} \]

### Step 8: Finalize the answer
Rounding off, the frequency heard by the crew on the aircraft carrier is approximately:

\[ f_o \approx 1996 \, \text{Hz} \]

### Conclusion
The frequency of the sound heard by the crew on the ship is approximately **1996 Hz**.

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