A stationary source generates 5.0 Hz water waves whose speed is 2.0 m/s. A boat is approaching the source at 10 m/s. The frequency of these waves, as observed by a person in the boat, is

Options

A. 25 Hz
B. 15 Hz
C. 20 Hz
D. 30 Hz

Answer

To solve the problem of finding the frequency of the waves as observed by a person in a boat approaching a stationary source, we will use the Doppler effect formula for waves. Here’s a step-by-step solution:

### Step 1: Identify the given values
- Frequency of the source (\( f_s \)) = 5.0 Hz
- Speed of the water waves (\( v \)) = 2.0 m/s
- Speed of the observer (boat) (\( v_o \)) = 10 m/s
- Speed of the source (\( v_s \)) = 0 m/s (since the source is stationary)

### Step 2: Write the Doppler effect formula
For a stationary source and a moving observer, the frequency observed (\( f' \)) can be calculated using the formula:
\[ f' = f_s \left( \frac{v + v_o}{v} \right) \]
Where:
- \( f' \) = frequency observed by the observer
- \( f_s \) = frequency of the source
- \( v \) = speed of the waves
- \( v_o \) = speed of the observer (positive when moving towards the source)

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:
\[ f' = 5.0 \, \text{Hz} \left( \frac{2.0 \, \text{m/s} + 10.0 \, \text{m/s}}{2.0 \, \text{m/s}} \right) \]

### Step 4: Calculate the observed frequency
First, calculate the numerator:
\[ 2.0 \, \text{m/s} + 10.0 \, \text{m/s} = 12.0 \, \text{m/s} \]
Now substitute this back into the equation:
\[ f' = 5.0 \, \text{Hz} \left( \frac{12.0 \, \text{m/s}}{2.0 \, \text{m/s}} \right) \]
Calculating the fraction:
\[ \frac{12.0}{2.0} = 6.0 \]
Now calculate \( f' \):
\[ f' = 5.0 \, \text{Hz} \times 6.0 = 30.0 \, \text{Hz} \]

### Conclusion
The frequency of the waves as observed by a person in the boat is **30.0 Hz**.

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