A body sends waves 100mm long through medium A and 0.25m long in medium B. If the velocity of waves in medium A is `80cms^(-1)`, calculate the velocity of waves in medium B. a) 1 m/s b) 2 m/s c) 3 m/s d) 4 m/s

Options

A. `1ms^(-1)`
B. `2ms^(-1)`
C. `3ms^(-1)`
D. `4ms^(-1)`

Explanation

To solve the problem, we will use the relationship between the speed of a wave, its wavelength, and its frequency. The fundamental wave equation is:

\[ v = f \cdot \lambda \]

Where:
- \( v \) is the speed of the wave,
- \( f \) is the frequency,
- \( \lambda \) is the wavelength.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Wavelength in medium A (\( \lambda_1 \)) = 100 mm = 0.1 m (convert mm to m by dividing by 1000)
- Wavelength in medium B (\( \lambda_2 \)) = 0.25 m
- Speed in medium A (\( v_1 \)) = 80 cm/s = 0.8 m/s (convert cm/s to m/s by dividing by 100)

2. **Use the Wave Equation:**
- For medium A:
\[ v_1 = f \cdot \lambda_1 \]
- For medium B:
\[ v_2 = f \cdot \lambda_2 \]

3. **Set Up the Ratio:**
- Since the frequency remains constant when the wave travels from one medium to another, we can set up the ratio of the speeds and wavelengths:
\[ \frac{v_2}{v_1} = \frac{\lambda_2}{\lambda_1} \]

4. **Rearranging the Equation:**
- Rearranging gives us:
\[ v_2 = v_1 \cdot \frac{\lambda_2}{\lambda_1} \]

5. **Substituting the Values:**
- Substitute \( v_1 = 0.8 \, \text{m/s} \), \( \lambda_2 = 0.25 \, \text{m} \), and \( \lambda_1 = 0.1 \, \text{m} \):
\[ v_2 = 0.8 \cdot \frac{0.25}{0.1} \]

6. **Calculating the Result:**
- Calculate \( \frac{0.25}{0.1} = 2.5 \):
\[ v_2 = 0.8 \cdot 2.5 = 2.0 \, \text{m/s} \]

7. **Conclusion:**
- The velocity of the waves in medium B is \( 2 \, \text{m/s} \).

### Final Answer:
The velocity of waves in medium B is **2 m/s** (Option b).