The tension in a string, obeying Hooke's law, is `x`. The speed of sound in the stretched string is `v`. If the tension in the string is increased to `1.5x`, the speed of sound will be

Options

A. `1.22 v`
B. `0.61 v`
C. `1.50 v`
D. `0.75 v`

Explanation

To solve the problem, we need to determine how the speed of sound in a stretched string changes when the tension is increased. We will use the relationship between the speed of sound, tension, and mass per unit length in the string.

### Step-by-step Solution:

1. **Understand the formula for the speed of sound in a stretched string**:
The speed of sound \( v \) in a stretched string is given by the formula:
\[ v = \sqrt{\frac{T}{\mu}} \]
where \( T \) is the tension in the string and \( \mu \) is the mass per unit length of the string.

2. **Initial conditions**:
We are given that the initial tension in the string is \( T_1 = x \) and the speed of sound is \( v_1 = v \). Thus, we can write:
\[ v = \sqrt{\frac{x}{\mu}} \]

3. **New tension**:
The tension in the string is increased to \( T_2 = 1.5x \). We need to find the new speed of sound \( v_2 \) when the tension is \( 1.5x \):
\[ v_2 = \sqrt{\frac{T_2}{\mu}} = \sqrt{\frac{1.5x}{\mu}} \]

4. **Relate the new speed to the initial speed**:
We can express \( v_2 \) in terms of \( v \):
\[ v_2 = \sqrt{\frac{1.5x}{\mu}} = \sqrt{1.5} \cdot \sqrt{\frac{x}{\mu}} = \sqrt{1.5} \cdot v \]

5. **Calculate \( \sqrt{1.5} \)**:
The value of \( \sqrt{1.5} \) is approximately \( 1.2247 \). Therefore, we can say:
\[ v_2 \approx 1.2247v \]

6. **Final result**:
Thus, the new speed of sound when the tension is increased to \( 1.5x \) is:
\[ v_2 \approx 1.22v \]

### Conclusion:
The speed of sound in the stretched string when the tension is increased to \( 1.5x \) is approximately \( 1.22v \).