Two bodies of different masses `m_(a)` and `m_(b)` are dropped from two different heights, viz, a and b. The ratio of time taken by the two to drop through these distance is:

Options

A. `a : b`
B. `m_(a)//m_(b) : b//a`
C. `sqrta : sqrtb`
D. `a^(2) : b^(2)`

Explanation

To solve the problem of finding the ratio of time taken by two bodies of different masses dropped from different heights, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We have two bodies with masses \( m_a \) and \( m_b \) dropped from heights \( h_a \) and \( h_b \) respectively. We need to find the ratio of the time taken by these bodies to fall through their respective heights.

2. **Use the Equation of Motion**: The distance fallen under the influence of gravity can be described by the equation:
\[ h = \frac{1}{2} g t^2 \]
where \( h \) is the height, \( g \) is the acceleration due to gravity, and \( t \) is the time taken to fall that height.

3. **Rearranging the Equation**: From the equation, we can express time \( t \) in terms of height \( h \):
\[ t = \sqrt{\frac{2h}{g}} \]

4. **Calculate Time for Each Body**:
- For the first body dropped from height \( h_a \):
\[ t_a = \sqrt{\frac{2h_a}{g}} \]
- For the second body dropped from height \( h_b \):
\[ t_b = \sqrt{\frac{2h_b}{g}} \]

5. **Find the Ratio of Times**:
- The ratio of the time taken by the two bodies is:
\[ \frac{t_a}{t_b} = \frac{\sqrt{\frac{2h_a}{g}}}{\sqrt{\frac{2h_b}{g}}} \]
- Simplifying this, we get:
\[ \frac{t_a}{t_b} = \sqrt{\frac{h_a}{h_b}} \]

6. **Conclusion**: Therefore, the ratio of the time taken by the two bodies to drop through their respective distances is:
\[ \frac{t_a}{t_b} = \sqrt{\frac{h_a}{h_b}} \]

### Final Answer:
The ratio of the time taken by the two bodies to drop through distances \( h_a \) and \( h_b \) is \( \sqrt{\frac{h_a}{h_b}} \).