two waves `y_1 = 2.5 sin100πt` and `y_2 = 2.5 sin102πt` ( where y is in meter and t is in second ) are traveling in same direction. the number of beat heard per second is

Options

A. one
B. two
C. three
D. four

Answer

To solve the problem, we need to determine the number of beats heard per second from the two given waves. The waves are given as:

1. \( y_1 = 2.5 \sin(100\pi t) \)
2. \( y_2 = 2.5 \sin(102\pi t) \)

### Step 1: Identify the angular frequencies
The angular frequency (\( \omega \)) of a wave is related to its frequency (\( f \)) by the equation:
\[ \omega = 2\pi f \]

From the equations of the waves, we can identify:
- For wave \( y_1 \), \( \omega_1 = 100\pi \)
- For wave \( y_2 \), \( \omega_2 = 102\pi \)

### Step 2: Calculate the frequencies
Using the relationship \( \omega = 2\pi f \), we can find the frequencies:
- For wave \( y_1 \):
\[ \omega_1 = 100\pi \implies f_1 = \frac{100\pi}{2\pi} = 50 \text{ Hz} \]

- For wave \( y_2 \):
\[ \omega_2 = 102\pi \implies f_2 = \frac{102\pi}{2\pi} = 51 \text{ Hz} \]

### Step 3: Determine the beat frequency
The beat frequency (\( f_{\text{beat}} \)) is given by the absolute difference between the two frequencies:
\[ f_{\text{beat}} = |f_2 - f_1| = |51 \text{ Hz} - 50 \text{ Hz}| = 1 \text{ Hz} \]

### Conclusion
The number of beats heard per second is:
\[ \boxed{1 \text{ beat per second}} \]