A man holding a tuning fork of frequency 90 hertz runs straight towards a stationary wall with velocity of 3.7 m/sec. The number of beats per second heard by him is (Speed of sound in air =340m/s.)

Options

A. 10
B. 20
C. zero
D. 2

Answer

To solve the problem, we need to determine the number of beats per second heard by the man holding the tuning fork as he runs towards a stationary wall. We will use the Doppler effect to find the apparent frequency he hears after the sound reflects off the wall.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Frequency of the tuning fork (f) = 90 Hz
- Speed of sound in air (v) = 340 m/s
- Velocity of the man (observer) (u) = 3.7 m/s

2. **Calculate the Apparent Frequency (f') when the man approaches the wall:**
The formula for the apparent frequency when the source is stationary and the observer is moving towards the source is:
\[ f' = f \left( \frac{v + u}{v} \right) \]
Here, since the wall is stationary, we treat the wall as a source of sound reflecting back to the observer.

Substituting the values:
\[ f' = 90 \left( \frac{340 + 3.7}{340} \right) \]

3. **Calculate the numerator:**
\[ 340 + 3.7 = 343.7 \]

4. **Calculate the apparent frequency:**
\[ f' = 90 \left( \frac{343.7}{340} \right) \approx 90 \times 1.010 \]
\[ f' \approx 90.9 \text{ Hz} \]

5. **Determine the frequency heard after reflection:**
After the sound reflects off the wall, the wall now acts as a source of sound. The man will now hear the frequency as:
\[ f'' = f' \left( \frac{v + u}{v - u} \right) \]
Here, since the wall is now the source and the man is still moving towards it, we use:
\[ f'' = 90.9 \left( \frac{340 + 3.7}{340 - 3.7} \right) \]

6. **Calculate the denominator:**
\[ 340 - 3.7 = 336.3 \]

7. **Calculate the new apparent frequency:**
\[ f'' = 90.9 \left( \frac{343.7}{336.3} \right) \approx 90.9 \times 1.022 \]
\[ f'' \approx 92.8 \text{ Hz} \]

8. **Calculate the number of beats per second:**
The number of beats per second is the absolute difference between the actual frequency and the frequency heard after reflection:
\[ \text{Beats} = |f'' - f| = |92.8 - 90| = 2.8 \text{ Hz} \]

### Final Answer:
The number of beats per second heard by the man is approximately **2.8 beats per second**.